I think you want an advanced degree and proper licensing to increase your odds of finding a competent T. A lot depends upon the issues you're having, how specific and serious they are, and finding a T with experience in treating those issues successfully. The fit is most important, but may not be readily apparent for a few sessions; the literature does show that the quality of the relationship is the factor that most correlates with successful outcomes, level of competence being equal.

One difference in the PhD or PsyD vs the MA/MSW etc are the requirements of personal therapy and supervision, but this highly depends upon the strength of the program awarding the degree. You should also find that a PhD in Counseling Psych vs Clinical Psych puts more emphasis on psychotherapy than research.

Also look to the environment in which the T practices: if the T is affiliated with an institution that has a good reputation, meets accreditation standards, etc, that can give you an assurance of competence that may or may not be as clear with a T in private practice only. Statistically the majority of abuse/ethical violation cases occur in solo private practices. There are lots of good Ts in private practice, but it's just something to be aware of.