I'd like recommendations on high quality camera preferably that can mount of helmets and can record the front and back. I saw some posts on other sites mentioning some crazy dudes just hitting people with their cars/car doors for fun even though they are in the bike lane. If I'm going to get hit I want to at least get some evidence of it and anything to identify them. Budget is $200 but prefer to be lower than that however if it's worth it I'd go over that amount.

Buy a used GoPro 3 White. They run about $100 and in 720 (30) mode their batteries will get almost 3 hours. Or you can spend $400 on a new GoPro where the batteries run for 1:45 mins.
If you want something very budget get the Polaroid Cube. Its cheap. I got mine for $77 and the image quality is VERY good.

You're more likely to encounter an inattentive or distracted driver than one who's deliberately aggressive. Offhand the only deliberately aggressive drivers I've encountered were two lanes away from me and just hollering or cussing, but not close enough to threaten me physically or for my riding to affect them. But the close calls have been from drivers distracted by their phones, inattentive or misjudging the relative distance and speed between my bike and their vehicles -- that's the usual reason for left or right hook cutoffs.

Most folks I encounter are very considerate and courteous. Those drivers far outnumber the handful of careless and rude drivers.

But I do usually ride with an Ion Speed Pro video camera to record incidents, especially if I happen to be injured and unconscious. Rarely does anything happen so I delete the videos when I get home. I've reported only one instance of reckless driving, when a Domino's delivery driver cut me off a few weeks ago -- the store manager and corporate safety officer responded satisfactorily so I didn't pursue it. Occasionally I'll record and keep entertaining videos -- group rides, recent holiday lights, etc.

For $200 you can buy one GoPro Session or two of several other action video cameras, including some that are discontinued or near end of cycle but still available: Contour Roam, Ion Speed Pro, Polaroid cube, Mobius, several knockoffs of the earlier GoPro Hero squarish camera design.

The two major 360 cameras cost well over $200. The Ricoh Theta is around $300 or more, and the resolution isn't quite good enough yet to record US license plate detail. The Nikon Keymission is around $500 and may have good enough resolution to record US license plates.

The future of action cams will be 360 types, but they haven't yet reached the sweet spot of satisfactory resolution at a price the market will bear.

Regarding the conventional video cameras that can be mounted on bikes, keep in mind a few factors:

• Some are designed more for action sports and close proximity, not for documenting traffic incidents and license plate detail. An ultra wide angle lens with nearly fisheye 170 degree field of view coverage at the typical moderate resolution will often struggle to record US license plate detail. A lens closer to 120-140 degree FOV will cover less overall but is more likely to record fine detail. Personally for traffic documentation I'd prefer a 60-90 degree FOV, equivalent to a 20-35mm lens on a full frame/35mm film camera. But a narrower FOV demands more precise aiming, which requires a rear LCD screen or laser aiming device (like the Contour Roam).
• Some are action cameras, some are dashboard cameras. There are some great values in low priced dash cams but they usually aren't weatherproof and may not hold up to the vibration and shock of bicycle riding. Weatherproof housings may be available at extra cost and bulk for dash cams.
• Most action cams will run 60-120 minutes on a single charge of the built-in battery, with 90 minutes being the current average. Some can be extended by plugging into an external USB battery. If you plan to ride longer than 90 minutes be sure the camera you choose can both run and charge off an external battery. Keep in mind that using an external battery may compromise the weatherproof integrity -- for example, my Ion Speed Pro is rainproof with the rear cap in place, but not when the cap is removed to access the rear USB port for charging/continuous recording while riding.
• Action and dash cams can be picky about recording media cards. Several models have customer complaints about glitches. The manufacturers will often suggest using only a name brand Class 10 or similarly compatible card from SanDisk, Lexar, etc. And beware of counterfeits, especially if a name brand top quality media card is selling for a much lower price than reliable vendors (check B&H, Adorama, etc., for guidelines to reasonable retail prices).
• If you ride at night you'll probably find most affordable action cams to have poor to mediocre image quality at night. The GoPros usually have better low light image quality. Some dash cams may have better nighttime quality, but again these may not be weather resistant or shock resistant without a protective housing.

At $200.00 you are looking more at entry level, name brand, than a quality spec'd out action camera.
I agree with what both previous posters have stated.
I would also like to mention my favorite brand, Sony. I much prefer the form factor and image stabilization of the Sony over the other brands I have compared it to, including Go Pro. Just never did like the look of that little square box mounted on top of helmet. I use the Sony side helmet mount on both my bicycle helmet and motorcycle helmet, just gives, IMHO, a neater appearance when I use a helmet mounted camera. Usually, while bicycling, I mount the camera directly to the bicycle mainly for simplicity sake. However, sound quality will be almost useless unless you spend some time making up some good anti vibration mounts. If sound quality is important, look for helmet/ headband mounts or the chest mount straps popular with the Go Pro's.
It is hard to get usable, detailed footage from a moving action camera. I have found that by using two action cameras, one mounted forward and one mounted rearward, it is a bit easier to get the details needed for a proper investigation. If you ride with traffic, which you should, the rear mounted camera will generate much more usable video than a forward mounted action camera. You will have the oncoming vehicle "insight/ in camera field of view" for a longer time which is needed to get the details. The front mounted camera can also get decent images/ video for objects/ obstacles that are in front of you. Mostly, I have found the rear mounted camera gets all the "detail/ money" shots while the front camera gets all the pretty scenery shots.
I usually will have my Sony mounted forward for the scenery shots while I have a Go Pro clone mounted for the rearward video. Both working together will usually produce excellent video documentation and very good detail down to the licence plate number.
I have also been looking at replacing my rear mounted Go Pro clone and while I would like to add another Sony 200 I very much have my eye on a Ion Air Pro (like Canklecat) which IMHO has a much better form fit than the Go Pro I have now. However, the Sony does have a very nice app that runs on my phone that can be used for up to 4 different action cams and you can actually watch what you are recording and more. It would be great if the Ion Air Pro would work on the same Sony app so I could do split single screen recording. But if it does not, it still sells for a lot less than the Sony 200 and I still may get one, the Ion Air Pro is definitely high on my list as my secondary action cam.
There are lots of choices out there, and from I have seen and read about, they are all pretty decent with different bells and whistles. If you want one for collision/ accident documentation I highly recommend a rearward facing camera. I also recommend getting the best media card you can afford, keep it clean by doing both a quick format followed up by a longer full format.
Don't expect any great videos no matter which direction the camera is pointed once the Sun goes down and car headlights come on.
Action cams can be fun, but for me personally I have never needed one to document an accident. I could have used one a couple of times to document aggressive driving but I choose not to.

I got a factory refurbished original gopro. The basic one, get about 4-5 hours of battery, video is pretty good happy with mine. Paid $99 on ebay with all the stuff it comes with in the box.

GoPro sells recertified cameras on an eBay store. They have the 3+ Silvers for $129 right now: GoPro Certified Refurbished Cameras | eBay Stores

That's a matter of opinion, experience, where you live, etc.

My experience with cameras is outside of biking, but if I cross the border and bike in wisconsin I get a lot more deliberate close calls than I do in minneapolis.

My opinion is that the thing is, with innattentive drivers I can ride defensively but that's about it. A helmet camera might help me get revenge or avoid increasing insurance premiums, but it doesn't really "protect" me.

With rude ******* drivers, my opinion is that the kind of people doing that behave significantly better if they know they're being recorded. Having a visible camera improves safety, because the ******* bullies (from obnoxious high school kids to adult pickup drivers) have often experienced that they can get away with things if no one else sees, but if there's video or an authority figure sees they get in trouble. They see a video camera and have the second throughts normal people do without it "gee maybe if this is on video I'll get in trouble later".

Your post echoes what I've been thinking recently. Often times security comes from the appearance of having security, e.g. roof rack locks which can be defeated with a screwdriver, cable locks which can be cut in 20 seconds with a pair of hand cutters, fake CCTV cameras, etc. While the method may not make it that much more difficult to carry out the crime, it discourages individuals from initiating it.

Are bike-mounted cameras all that visible to the drivers in question? In that case, what is the best camera that clearly says "I'm recording you so please think twice about your behavior"? Something helmet-mounted and well-recognizable, like a GoPro? Something with a blinking bright red light?

Honestly, no, bike mounted cameras are not visible. Red lights are easily confused with tail lights.

The only remotely visible camera is a helmet cam. The main challenge of a GoPro is the battery life isn't there. Realistically, you need a camera with 4+ hr battery life.

I recently had a passenger tap me on the butt while the car drove by. Unfortunately, my camera was off because it was a wide open street with minimal traffic. If the camera doesn't have enough battery to run for the entire length of the ride, it's not that useful.

Samsung 360 camera is in the price range and can get at Amazon. It's water resistant, but not water proof, so rainy days would/could be an issue.

- Cpt. Bob

Wow, wouldn't that qualify as assault/sexual assault? Probably not worth reporting, though if the local PD was bike-friendly and it was easy to turn the footage over (if any) it'd be worth a shot just to generate a warning.

My GoPro Hero 3+ can, if you turn it on and off between shots. I use mine with the battery back pack, and get just about a full day of filming out of it doing that.

Sorting through 12 hours of footage is incredibly tedious anyhow, I prefer just to have the interesting stuff.

I think you need to rely on a jersey or vest if you want to get that message across. I'm pretty sure I've seen a high visibility jersey with a picture of a British style road safety camera on the back and I've definitely seen a high-viz vest with that picture and the word "FILMING" on it.

What about Cycliq Fly12?
https://cycliq.com/

10 hours of recording. But the camera deletes old files when there is no space. That's the main advantage over GoPros, etc.

I don't expect that you get anything better than it for the same money.
These cameras all use the same image sensors from Sony, etc.

I find it useful to have a camera-light combo at night. It helps to record license plates.

If you're willing to do a bit of assembly - yes - sort of. There are some very small cameras used by r/c modelers and others that can be hacked with weatherproof housings and external battery packs. I'd imagine with a large enough USB cellphone external battery you could get the life you are looking for.
As for the quality of the camera - in particular license plate details - please do some research as it has been awhile since I've looked at them. One brand that seems to have a lot of online support is Mobius but I'm sure there are others.