Does NP-completeness require to find the solution?
You are right that NP-completeness applies only to decision problems. What they mean by "Problem 2.1 is NP-complete" can be either


  The decision problem corresponding to Problem 2.1 is NP-complete


or


  The decision problem corresponding to Problem 2.1 is NP-complete, and the search problem is in FNP


Here the search problem is finding a clique of size $k$ if one exists. We think of the search problem as a two-place predicate $f(\langle G,k \rangle,K)$ which is true if $G$ is a graph and $K$ is a $k$-clique of $G$. This search problem is in FNP if the predicate $f$ is in P. (Usually FNP is defined for functions, i.e. for relations which are functions, but you can also define it for general relations.)