"Chess problem" is a term of art that refers to an artificial composed position with a unique solution that is constructed to both be a challenge to the solver and have aesthetic value. They often have constraints on the solution such as that White must deliver checkmate in two moves (three ply). This is what I assume you're referring to.
A position from an actual game (or that easily could have been) that demonstrates a tactic (or combination of them) is generally known as a "chess puzzle", largely because the term "chess problem" was already squatted on.
Somewhere in between the two is the "study", which is a constructed position, less artificial than a chess problem but still very carefully made to have a unique solution that walks a tightrope and generally requires absolutely exact calculation rather than working by general tactical principles.