These are my Xbox recommendations:
Black
is my all time favorite FPS. It's on both the PS2 & Xbox. (I played the xbox version.) Awesome explosions, awesome sound, & awesome particle effects. A lot of people get frustrated because they think it takes 30 bullets to kill a guy, but they don't realized the bad guys get some recovery time after each bullet. (i.e. they're invulnerable for a moment as they recoil from getting shot.) If you just shoot, pause, shoot, pause, shoot, you can kill them with just a few shots. Beat the game on normal difficulty, & you unlock unlimited ammo for all guns... including rocket launchers. =D
Burnout 3
It's a lot like Burnout Revenge which is available for both the Xbox & 360. Both have the best sense of intense speed I've ever experienced in a game. The biggest difference between Burnout 3 & Revenge is Revenge lets you smash all traffic off the road, where Burnout 3 only lets you smash into other racers w/o crashing.
Crimson Skies
Is a great flying game. Mechanics/controls are simple, graphics are great, story is good. I've flown around quite a bit just to enjoy the scenery. It also has 4 player multiplayer matches.
You might want Halo 1 or 2.
They're the only Xbox 1 games to get special BC treatment so they're natively rendered at 720p on the 360. All other BC games are only rendered at 480p & scaled up to 720p or 1080. Personally, Halo 1 bored me, but Halo 2 not only held my interest, but it looks great in 720p.