Given the fact that saving is completely broken, only one out of the four planned characters is playable, the game ends after the first boss, and multiplayer is almost entirely non-functional, I'd call it Alpha.
TBH, though, Alpha vs. Beta is semantics, as it can vary(See Minecraft, which was in public Alpha for years). Either way, though, I'd still call it a private beta at best. I guess this gives some credence to your theory about JJ3D being leaked by WTG, since it would make sense for them to tag "Public beta test" onto the setup if that was the case.

And speaking of...
That does sound a lot more plausible, especially when you consider that Cliff Bleszinski showed interest in it being spread around back when that was still being talked about, and what I just said about the setup thing.

Oh yeah, I remember that. XD No idea what they were thinking there.

Anyway, might aswell post another fact: Jazz 2 was originally to have two different opening cinematics; one for the demo, one for the full game. The demo one is the one you see in-game currently.
Dean Dodrill(Who animated the demo one aswell) started work on the full game version, but never finished it. What he did finish of it can be seen here.