Your Problem 2 is indeed a promise problem. It is not in NP, but if you like, you can say that it's in a class Promise-NP. (This term doesn't seem very common, though, unlike the other class "Promise-BPP".)

See Oded Goldreich's survey [*On Promise Problems*](http://www.wisdom.weizmann.ac.il/~oded/prpr.html). The promise classes are different, and sometimes much stronger, than their plain versions. Among other things, a problem in the promise version of NP∩co-NP need not be in NP, and (as commented [here](http://blog.computationalcomplexity.org/2006/07/definitions-of-advice.html#115259881395001905)) there are NP-hard problems in promise-NP∩co-NP while this is not believed to be the case for NP∩co-NP.