Is the term "payload" used exclusively for the payload of (let's say
or it may be used to describe the useful information in
the response of a request?
Yes, that's a valid use of the term.
You distinguish between the two by how the term was used in a sentence; if it was used unclearly, then request clarification.
If you ask "What's the server interaction look like for transmitting a payment?" and someone answers "The payload is XML", you can likely presume that they mean that both the request and the response will be XML (or possibly that the format of the response will match what the headers say will be requested, or will match the format of the payload of the request). If you don't want to make that assumption, you would probably ask "The request payload, the response payload, or both?"