Dragonball Z: Ultimate Battle 22 (PS1)
Dragonball Z: Shinbutohden (Saturn)
Dragonball Z: Budokai 2 (PS2/GCN)
Dragonball Z: Budokai 3 (PS2)
Dragonball Z: Sagas (PS2/GCN/XBox)
Dragonball GT: Final Bout (PS1)
I believe I like the intro to Budokai 2 or Final Bout the best. Funny to say that the weakest of titles has one of the best intro animations, but it is true.
Sagas had a very interesting intro that showcased a full look back at Goku's early battles. It was a nice touch to an otherwise mediocre game.
Final Bout's intro is cool, the only cool part about that game if you are honest with yourself. Showing Piccolo fighting Perfect Cell always seemed odd to me, but aside from that it was rather impressive.
Ultimate Battle 22 has the superior intro, even though the comparable Shinbutohden is the better game overall. It is a nice run down the roster of characters, but not much else to it.
Budokai 3 is by far the best of the Budokai series (In my opinion of course), but its intro, while animated beautifully, has too many odd aspects to it. Super Saiyan Gohan never fought Meta-Cooler, Gotenks Never fought Kid Buu, and even if they are hypotheticals, which is fair enough I guess, There is no way in heck Gogeta would take a blow from the likes of Broly and bleed and be knocked back half a mile. That one bit just kinda annoys me when I watch the intro. Good thing they include the Budokai 2 Intro as an unlockable, that redeems that one shortcoming.

Budokai 2 has the best intro in my opinion. It seems to capture the overall dragonball impression one would expect from an intro. It shows many story accurate scenes and battles and much of the supporting cast. I think I enjoy the intro music on it too the best of them all.
I am positive many of you may disagree with me, and in this discusion, I quite frankly welcome it. This isn't anything other than a leisurely look at these animated segments unique to the world of dragonball video games.