What are the different components of degree titles called?
To take one of your examples:

Master of Arts in Psychology: Behavioral Health
|_A__|_B_____|_C___________|_D________________|


"Master" is the degree. Etymologically, "degree" comes from "degré": the step of a stair (or  rung of a ladder). "Degree" can be synonymous with "level" or "extent" - it is a measure. You could say "the level of a degree", but that would be a tautology. It's not incorrect, but "level" is redundant there - "degree" already says everything that you'd be trying to get from "level". Degree is a very flexible word, and can be used to refer to just part A, parts A+B, parts A+B+C, and parts A+B+C+D.

Then either:
"Psychology" is the subject and "Behavioral Health" is the speciality, or "Psychology: Behavioral Health" is the subject. You might also hear "field" used instead of "subject".

"of Arts" doesn't really have a well-recognised name. You could call it type, kind, field, sort, school or faculty, but in the end, if you want to ask what sort of Master's someone has, you'll probably end up saying something like: "What kind of Master's? Master of Science, Arts, MBA, something else?"