We all are good friends (or at least "friends"
here). Sometimes our understanding of things and terminology usage is broad in meaning, and sometimes very specific.
Turning Between Centers has in its fundamental process - the exclusion of mandrel usage. If I had not seen your photo, and you had a question about a problem, I and dozens of others would have been answering from the prospective of "not using a mandrel", and there would probably been some confusion in the answers.
TBC came along looong before PSI came out with their "mandrel saver", which lets people use adjustable length mandrel spacing. It gives rise to the idea that it is similar to TBC. But TBC is without mandrel, and between the live center and drive/dead center. TBC can be used with bushings and without bushings, but TBC does not mean "without bushings" in and of itself.
I like your setup and it will help in keeping CA from sticking to bushings, since you didn't use bushings in that set up.
TBC = Turning Between Centers directly on the head stock dead drive and tail stock live center.
Bushings are optional.