I had a Betta (named Finger) that, when I first got him, was in a 2.5 gallon tank. He would just sit in the corner under a plant and not move, he would also be very fearful of anything going on outside of the tank. Well I moved him into a 1 gallon and he suddenly had a personality! He would swim around everywhere, pick up rocks, flare at people passing by the tank and even swim into the filter just to play in it. :)
He is still in the 1 gallon, he just lives with my mom now.
I think it depends on the Betta's personality, I would NOT use my story as proof that Bettas should live in small tanks, it is just that Finger likes small tanks more.
Just as one of my other Bettas, Melvin, likes larger tanks more. He was at first in a 1 gallon and he was dull and slow moving, then I put him in a 5 gallon and now he is one of my most lively Bettas.
So yeah, it depends on the Betta. But yes, I do think some Bettas like smaller tanks. :)
Bettas - Giovanni (HM), Burly (HMPK Giant), Potato (EEHM), Vesper (HMPK).