Let's say I have a fork of the blockchain into chains A and B. My client currently only sees A and I submit a transaction.

Would it now be possible that someone else also applies the same transaction to B? Or is the transaction somehow bound to the chain (e.g. by specifying the hash of a chain block that must exist before the transaction is considered valid), so that a transaction submitted for fork A is invalid for fork B.

The reason why I'm asking is that some "Bitcoin banks" seem to only credit received coins after a given number of blocks have passed. If it would be possible to bind a transaction to a specific block chain (e.g., by specifying a required "previous" block) one could avoid this problem, by instantly crediting received coins, but by binding newly sent coins to the same fork of the chain where the previous coins have been received.