I'll need new tires for my touring bike, a 1993 bridgestone xo-2. The tire size needs to be right around 26 x 1.5 or 1.6. I'm mounting fenders on the bike and that's the tire size I'll need with fenders.

I'm considering two tires: (1) schwalbe marathon supreme touring 26 x 1.6; (2) panaracer ribmo 26 x 1.5. Any other tires I should be considering which you would recommend? Any issues with using these tires? I'm leaning towards the schwalbe but it's expensive so I'm looking at alternatives.

I have been quite happy with the plain Schwalbe Marathon (with GreenGuard) in 26X1.5 (or 559X40mm). Wire bead, not too expensive.

I also bought some Vittoria Randonneur Pro folding bead tires in the same size to carry as a spare, but I have not installed them so I can't offer an opinion on them. At this price I might buy one or two more. If you keep your quantity down, the shipping cost is reasonable, put one more thing in the cart and shipping skyrockets.
https://www.planet-x-usa.com/i/q/TYVT...o-folding-tyre

Brifters on an old Bridestone?

Yeah, the brifters are a little heretical I know. I may end up going with bar end shifters when I'm done with this bike.

I have Ribmo and they have worked well for several thousand miles. No flats yet. I also have Schwable tires, though not the model you seek, and they too work well.

I just got a pair of 26X1.5 Marathon Racers (HS 366). Easier than expected to mount. Only have a few hundred miles on them and so far they are nice. I don't do dirt but do live on an unpaved hard packed dirt/gravel road they work well with caution. I came from some cheapo 1.25 slicks.

I have never cared for the ride of any of the Marathon incarnations that I have tried. The Compass tires I've been rolling the past year or so have been the best 26" tires I have ever used. Don't pump them up to the recommended max. pressure; just pump them firm enough to not have sidewall issues when riding. (That may not be an issue on a half-bike, but the handling definitely improves when we keep the pressure down around 50 psi on our tandem.) We get about 4000 miles per set on our tandem with a 300 pound team, so they definitely give you your money's worth.

Compass Bicycles: 26" Tires

i've been satisfied with these CONTI Travel Contact tires. I think I've had them about 3 years - lots of dirt, gravel roads. Small bit of loaded touring. More to come!
https://www.conti-online.com/www/bic...ontact_en.html

My LBS sold me these SERFAS VIDAL tires - 700 x 38.
Only about three rides so far. Didn't get sliced up on gravel. Ok on dirt. Field crossings about what you'd expect at 80psi.
A little noisy on pavement, but cheaper than most tires. Initially, I think they might be OK.

I love those Schwalbe Marathon Supreme (20x1.6) on my Bike Friday Pocket Llama. Comftable, good grip, durable and excellent puncture resistance

I agree with Carfree.
Do yourself a favor and spend the extra money for the Compass 26” x 1.75” (42 mm on my rims).
They are fast and comfortable on the road and do fine on dirt and gravel.
I tour on them with a weight of 215lbs (bike+gear+me) and set the pressure at 38psi front and 48psi rear.

What is your loaded weight?

I am a bit under 350, and on my PDG 70 series, I run a Kenda Kwest K193 100 psi 40-559 on the rear, and a Nashbar City Slick 100 psi 32-559 on the front, both on Velocity Aeroheat (18mm inside width) rims. I have over 1000 miles on this combination - no issues at all, just pump them up once a week. They both still have mold flash in the tread area.

I've been using Schwalbe Marathon (Greenguard most recently) for the last four years on summer tours averaging 1000 miles per summer. Before that I had Ritchey Inverts, both similar in size at 26 x 1.5. I liked the Ritcheys best for their tread pattern and long life but those are not made anymore. The Marathons seem to have a softer rubber so they didn't last as long. I like the Marathons' range of acceptable pressures (100 psi max). You can play with the pressure depending upon the surfaces you are riding on. On any given tour I'm on a wide range of surfaces so that is important to me. Puncture resistance seemed equal between the two but I'm comparing the lower shelf Marathons with the Ritcheys. I have Greenguards now and expect they will outperform both of those in the long run. Can't speak to the wear characteristics of the top shelf Marathons. I got both pairs of the Ms through Tree Fort Bikes.

Total tire noob. My 26" currently is running on a 1.9 front and 2.125 rear knobbies. I see the compass tires are listed as "clinchers.". Does that require a different type of rim?

No, it means they aren't tubular racing roadie type tires. You're fine.