Nonuniform input distributions in average case analysis
Yes, the expected running time under some other distribution would still count as an example of average-case analysis.  However, when you describe it to someone, make sure you explain what distribution you're using.  I wouldn't recommend that you just call it "average-case analysis" without also explaining that you're using a non-standard probability distribution.

The worst-case distribution is always a point distribution that assigns all its probability to a single input.  In  other words, the worst-case probability distribution that makes the expected running time as large as possible is in fact just a single input: the input that makes the algorithm run as long as possible.  Consequently, this kind of worst-case analysis coincides with the "standard" notion of worst-case running time.  For this reason, there is no need for a special "name" for this; worst-case running time already covers it.

The same is true for best-case running time and the probability distribution that makes the expected running time as small as possible.