Two different Links or not, I think Miyamoto intended them to be the same at their very core.
In reality Link probably wouldn't have been helpless as a rabbit-person. I think it was just a stipulation in the game since it clearly makes no sense to be unable to use
anything in that form. But that's just my opinion. I mean, as the argument stands, the wolf can't wield a sword, either. The wolf has teeth, claws and can use its weight more efficiently for attacks, but that doesn't make the bunny helpless. Well, if you disregard the stipulation in the game, anyway.
On the other hand, if being a pink rabbit reflected his heart, I don't see what would be so embarrassing about it.
He should pick to be a faery. Mwa-ha-ha!
And I don't see how he could be timid. You don't see timid people having courageous adventures so willingly.