I think this means that your T needs to do supervision with a T-in-training, that's what student supervision is. Which means that student-in-training has a few clients and needs to be supervised with those clients. So the student's supervision time needs to be arranged around his or her client's schedule, your T's schedule, and the student's other scheduling commitments. That's a lot of scheduling to accommodate, much more complex than just individual clients' schedules.

At least that's what "student supervision" meant to my wife, when she was in graduate school for clinical psychology. (She now does research in that field, not therapy). Maybe it doesn't matter to you if you've been asked to switch because your T needs to meet with his student for supervision (and he has probably had a long standing commitment to supervise students) as opposed to being asked to switch for another client, who is a student. But if your sibling rivalry buttons are being pushed because you think that T is favoring another client over you, as opposed to needing that time to supervise another T (in training), then maybe this helps.

It seems to me that no matter what is distressing/triggering you, it would make sense to work through this issue. It seems that your interpretation about what this situation means is really at the heart of the problem. It's not unreasonable for your T or the receptionist to ASK you to switch-- and the word "need" doesn't mean that you are being forced to, people use this word all the time to communicate all kinds of things, like "it's really important to us that this work out, we're not asking this for a trivial reason". And just because someone asked you to switch your day or time, doesn't mean that they are saying you are less important than anyone else.

Why shouldn't you be thoughtful and considerate of your T's schedule, if it is no skin off your nose? Perhaps there are other better ways to "take back your power" than making a fuss about this. If one time is reasonably equivalent to another-- and I see my T at a different time and day almost every week-- so what?

I just don't think this requires as much energy as you are putting into it. If you just don't want to do it, then call back and say you can't make that work with your schedule. You don't need to explain why or pretend that you're sorry, you can just be straight. And then talk to your T about what you heard and what you think it meant to you, and work through it.

My advice is worth what you paid for it.