Does the Timestamp of a Block Affect the Deterministic Output of Its Hash?

Yes, the timestamp is one of the data fields included in the block header. Since the block header is the input to the SHA-256 hash function, any change to the timestamp will result in a completely different, deterministic hash output.

The timestamp is a variable element that changes from block to block, which is necessary because the nonce alone might not be enough to find a valid hash within the required time.