I'm looking for a formal and broad definition.
Intuitively, I would say that a problem where various checks have to be done is a search problem. Is this the only criteria? How can we know that a problem requires multiple checks?
Alternatively, I also think intuitively of problems with multiple paths as a search problem. In this case, only a single check might be necessary but multiple comparisons are needed in order to know which path is the best to follow.