Yep 360 for sure. DIRT is awesome for PC but requires such a good system that I'd want it for 360. Forza on Xbox 1 was the best racing game on Xbox, so Forza 2 on 360 can only be that much better.
The price difference is huge, but racing games in HD on a big TV would be awesome, and if you plan to keep this for another couple of years, you'll be disappointed with the lower resolution graphics and the lack of new releases on the Xbox. I like the Xbox the most of last gen, and it has a lot of good games, but nothing new is coming out.
I'd start off with a 360 premium (or core with HD), racing wheel, Forza 2, and DiRT. That'll set you back a lot I know, but look for used deals on the wheel and games maybe and wait a bit for a price drop or just get a used 360 if the warranty can be transferred (doubt it though). Pick up used Xbox 1 games cheap after checking the backwards compatibility list. PGR2 on the Xbox 1 was good but you can probably find PGR3 for the 360 for $20 or less. Also keep an eye out for a cheap used copy of Test Drive Unlimited, mostly if you are gonna have Live. And there are those crappy Burnout games if you're into that. Finally, there are NFS games for Xbox and 360, so look into used copies of those.
I'm jealous... I wish I had a big HDTV, 360, wheel, and 4-5 of the best Xbox and Xbox 360 racing games!!!
PS2... not a chance. That thing sucks. And the controller is crap for racing compared to analog triggers on the Xbox controllers.