You are right, I think, that there are different reasons for appellations in different situations. The "power differential" reason for using "client," however, has never made sense to me (but then, I am decidedly against political correctness for its own sake). Of course there is a difference in power, which varies with the situation and patient, and which is not always in the direction of the therapist. The therapist's special knowledge and expertise, experience with patients, and at times even his/her authoritative role are routinely important in helping the patient. To say that a "power differential" is always bad is illogical.

Most people don't come to serious therapy to interact with an "equal" or a "friend." Patients really do want (and need) someone who is much better than they at solving the problem at hand (no matter what the depth of the problem or style of the therapy). That may include an ability to see things the patient can't see in himself/herself (which can sometimes be done with a mere friend, but not usually as well), but the additional expectation (by the patient) that the therapist is "powerful" enough to guide the patient to solve them is important. Respect is very important, as you point out; paying lots of attention to whether or not there is a "power differential" in the therapy setting seems silly to me.

Tom Szasz once told me in Syracuse, now some 20 years ago, that all he did with his analytic patients was "schmooz." I don't believe for a moment that that's enough.