Traceability in Pure Testing Projects

I have a question about addressing requirements traceability for pure testing projects (Understanding requirements->Writing Manual Test cases->executing them). If the application is not developed by us, what information other than Module name, Requirement ID, description, and Manual Test Case ID needs to be mapped?

There is no definitive answer to your question. The actual answer is up to you and your organization to decide what is necessary for your traceability. What does each testing project need to know about traceability? If you can answer that question, then you have the answer to your question as well. What you have listed sounds reasonable, but only you can determine if it is complete or that you need to add other elements.