It is very common to define a decision problem $L$ in the following way. Let $f \colon \Sigma^{*} \to \{0,1\}$. Then $L = \{x \in \Sigma^* \mid f(x) = 1\}$. Effectively, $L$ contains all instances $x \in \Sigma^*$ that have a "yes" answer.
Are there similar concise definitions for other kinds of computational problems, like function problems, search problems, and counting problems?
Since a function problem is a generalization of a decision problem, I suppose we might take something like $f \colon \Sigma^* \to \Sigma^*$ and let $L = \{(x,f(x)) \mid f(x) \text{ exists}\}$, but that feels kind of hand-wavy. Search problems are like function problems with multiple instance-solution pairs, so we could use almost the same definition, and then counting problems can just be defined as the cardinality of the search problem set.
