What type of commitment scheme is it?
My first thought was that it is perfectly binding.


It's not.


  $commit_B(C_{H1}) = commit_B(C_{H2})$


That's fine


  This would be a contradiction, because $commit_B$ is perfectly binding.


Actually, that's not a contradiction, because it's possible that $C_{H1} = C_{H2}$

After all, $H$ is not perfectly binding; what that means is that you can have two different messages with the same commitment (one would hope that it is hard to find both sets of the random coins that would yield the common commitment; but we're talking about existence, not computational complexity).  In fact, if $H$ is perfectly hiding, then any valid commitment could possibly correspond to any message (if there's a message that it could not reveal, then the receiver could deduce that message was not the committed one).

Hence, $commit_H(M_1) = commit_H(M_2)$ is perfectly possible, given the appropriate random coins.

And, so, when we open the larger scheme, the committer can (given unbounded computation) change his mind what he opens $H$, and so this is not perfectly binding.