Whenever I hear or read the phrase "bitcoin client", I assume the person is talking about a full node. Perhaps that's just a left over artifact in my head from previously referring to Bitcoin Core as the Satoshi Client.
In p2p network every full node can be viewed ether as the client ( when it leeches the data from surroundings) or the server when it provides them on the request. Gradually the term "client" became distanced from its original meaning and people often use it as a synonym of full node though it is not quite correct. I think would be better to leave the words with their original meaning.