'Without prejudice to damages' meaning

without prejudice TFD

By invoking this statement, a party is asking for assurances from the other party that the information provided by them will not be used against them to their disadvantage. Without any loss or waiver of rights or privileges.

If I'm understanding it correctly, in other words, terminating the contract does not bar X from claiming damages as well?

Yes, you are correct.

X would be able to terminate the contract (which is inherently reciprocal), but still retain the faculty to claim for damages. Without doing so, X would still be liable to comply with his/her part of the contract, all things being equal.

"Without prejudice to your right to damages" might mean "without having any adverse effect on your right to claim damages" but if you were drafting a contract you should consider the possibility that it could be argued that the AMOUNT OF your damages was affected despite such a clause.

That's why we go to law school.