It's true that the rent payment depends on the previous transaction's signature, but if that payment (even if mutated) has made it into even one block, the wallet software that generates the rent transaction is able to figure out which transaction on the blockchain is responsible for the change it intends to use.
That's correct. You should probably wait a block or two for safety, or more for really sensitive things, but regardless the main point here stands: because of this you cannot safely immediately spend the output of a transaction. That hasn't been a warning in essentially anything I've seen.
