Your Problem 2 is indeed a promise problem. By definition, it is not in NP, because NP is a class of decision problems, not promise problems. But if you like, you can say that it's in a class Promise-NP. See Wikipedia or Oded Goldreich's survey On Promise Problems.
For a decision problem, all strings are either YES instances or NO instances. For a promise problem, some strings are allowed to be "invalid": neither YES nor NO.
In general, if you have a nondeterministic algorithm for a promise problem, then when you feed it inputs that don't satisfy the promise, it will either accept or not — so the the sets of strings accepted and not accepted by your predicate will be supersets of the actual YES and NO instances respectively (and have intersection with the "invalid" instances). In this particular problem, because your condition is a necessary one for 3-colorability, the set of accepted strings will be exactly the set of 3-colorable graphs, but its complement will include graphs that don't satisfy the condition. (If you like, you can artificially change the problem to a decision problem with the same set of YES instances, but then your Problem 2 becomes the same as Problem 1 and therefore in NP.)