Hello Andrew,
Any labels I can offer will also be subject to debate of course since I don't have any official word from Breguet, but I think these changes would make it more easily understood by most watchmakers at least:
"main wheel" = barrel, the toothed portion of the barrel is sometimes referred to independently of the barrel in some manner, but most frequently the barrel is referred to as a single piece. The components of the barrel might be the barrel drum, the barrel cover, barrel arbor and mainspring
"transfer wheel" - I'm not sure there is a good consistent way to refer to this type of wheel, but transfer wheel may be as good as any. In the article I wrote about the Union Glashuette Julius Bergter Small Seconds, I called the pinionless wheel in that power train the "second intermediate wheel", which is surely not the best thing to call it by any stretch. Intermediate wheel is generally a safe thing to call just about any wheel between two others without a clearly identified name however, and transfer wheel sounds good to me too.
"centre wheel" - I can't go there for this wheel for obvious reasons. Some manufacturers used to refer to this type of wheel as an "offset center wheel", ETA refers to it as an "intermediate wheel" (clever) and most modern manufacturers refer to it as a "great wheel" (the likelihood of it being confused with a great wheel in a fusee power train is minimal in this day and age). No idea what Breguet calls it.
"third wheel" - Agreed.
"fourth wheel" - This is clear to most English speaking watchmakers, although whether it is British or American nomencalture originally is unclear to me. Most Europeans refer to it as the "seconds wheel" or even "second wheel", which can be very confusing to us yanks, where "second wheel" very often meant "center wheel" (great wheel), being the second wheel in the train after the barrel. It's possible our European brethren are counting the escape wheel as the first wheel in the train, but more likely they're referring only to the fact that this wheel generally carries the second(s) hand. In this instance, it does not carry the second hand and may or may not turn once per minute, but I think I'd still call it a seconds wheel. Fourth wheel is not completely unclear though, as mentioned above.
"spring stud" - This label is pointing to the movable stud carrier, which of course carries the stud itself. Breguet appears to be using a Geneva stud-style mecahnism here a la Patek, Vacheron, etc. The "movable stud carrier" label is actually pointing to the locking screw for the movable stud carrier.
"intermedaite wheel" - This term is generally safe as I mentioned above, but also not very descriptive unless the manufacturer has used it on some technical documents and it's the only intermediate wheel identified on the movement. This is probably a setting wheel of some sort (used to set the hands), and in such case "intermediate setting wheel" would be more descriptive, but I really don't know if it would be correct. Barring the use of Breguet's actual terminology for this wheel, I would resist referring to it at all if possible, to the point of pretending it is not really there or, if pressed, being as non-technical as possible, "You know, that little wheel to the right of the dial!"
Regarding the picture of the back of the movement, I think the vast majority of the labels are more descriptive and helpful than confusing, with the exception of "crown wheel". When more than one wheel is present between the winding pinion and the ratchet wheel, only the first wheel is generally referred to as a crown wheel, all others being "intermedaite crown wheels" or "winding wheels" or something of that sort. If Breguet does in fact refer to it as the crown wheel, then so be it.
"detent spring" - More commonly referred to as setting lever jumper, but to American and English watchmakers detent spring is clear too.
Thanks for taking the time to create these labelled pictures. I'm sure many Tradition owners will get a lot of pleasure out of learning what to call some of the bits they can see through the front and back of their watches.
_john