It sounds like this debate stems out of some national chauvinism wherein your friend wants to score some historical points for the Poles over the Russians. The actual answer depends on your perspective; Poland-Lithuania was a commonwealth with a unified government, not merely a personal union, so it would be strictly correct to refer to the soldiers of this polity as Polish-Lithuanian. If you're more concerned with their ethnic identities, you could certainly refer to them as Polish, Lithuanian, Ukrainian, etc.

To be absolutely precise the war should be called Polish-Lithuanian-Russian war. Calling XVIIth century Russia "Muscovy" is so XVth century and actually is a part of a historical myth, so we should not follow it.

In terms of official names, as it already has been said, the forces of the Commonwealth should be referred to as "Polish-Lithuanian", as the name of the state suggests. But your friend might be referring not to the official name but rather dynamics within the Commonwealth that subsequently led to the domination of Polish nobility, culture and language within the state. I'm not knowledgeable enough of Lithuanian history, but as far as I know, in years 1605–18 Lithuanian culture was still influential enough to be respected as an "equal" part of the Commonwealth, so even from this point of view its forces should be called "Polish-Lithuanian forces".

Referring to Polish-Lithuanian Commonwealth as simply "Poles" comes from both shortening and the future of that state, in which Lithuanian part became strongly dominated by the Polish one.

Please, do note that there is a lot of historical mythology, so every argument about these events even in the naming of things is heavily influenced by the myths. Your friend might be a victim of that, I might be a victim of that, wiki pages are 100% victims of that, so the safest way is to follow official names of the states.