I've started looking for another bike to use primarily as a winter trainer and possibly to run some errands into town (approximately 10 miles) which means it would have toe clips instead of clipless pedals. I am leaning toward a vintage/classic with Raleigh - first, Fuji - second and Trek - third, but I can also be swayed. What would you recommend I look for that is within a reasonable price - preferrably $150-$200?

When the time comes when I ask myself the same thing its pretty much whatever is at the dumps recycle center I can buy and fix up if I dont have anything too use already. I cant use anything too nice cause winter is actually winter where I live, not 10 degrees cooler and a tad more rain.

Where do you live? Some areas are a lot cheaper to buy in then others.
How do you plan to acquire this bike? Craig's List? Ebay? or a friend?
What sort of drive train are you looking for? I assume downtube shifters
are in the plans, but are you looking for index or friction?

I'm in central WA state. I'll more than likely be doing this via Craigslist or visits to the LBS to see if they have older used bikes. The one in my town told me he has an older Trek with 27 in tires but has to remember who brought it in before selling it. I don't know a lot about the different models and would prefer one that isn't much heavier than the mid-20 lb range. I'm not real picky on the drive train or shifters but definitely would like downtube shifters - if not just for the nostalgia aspect. I do prefer something where parts won't be a big issue (i.e. available and not cost prohibitive).

I'm not too good at discerning what are people needs are, but you can't go too far wrong with a Trek. Even the 330s aren't too bad. Can't speak for Fuji's.
I have two Raleigh Gran Prix, plus a Grand Sport, and a Competition. The
Grand Sport and Competition have butted 531 frames. The Gran Prix will get
me around and I don't worry as much about locking them up for a few hours
in downtown Boston. How much biking in snow and ice do you plan to
do? I don't do much myself. Tires can make a difference in snow, but you
can go down with any sort of tire if the roads are slick.

The Competition is in the low twenties, the Grand Sport and Gran Prix are north of 25 pounds.
Most mid and low end Trek Road bikes I believe come in under 25 pounds.

I would go for a Fuji, but im partial. If you want good components/ light weight go for a Team or higher