Cat amongst the pigeons !
Although I have heard it many times before, I still am yet to see it written in stone that an M1 visa is REQUIRED to conduct flight training on an ad-hoc basis (Part 61 for ex) under 18 hrs per week.
Sure, a valid visa IS required to enter the USA or if you qualify for the Visa Waiver program this is in leiu of a visa. You are then in the USA on a valid B2 visa (you are stamped in B2 in other words).
This is not advice by the way, just an observation. Ask any US Pt 141 school and they will tell you you need an M1 visa, no question, probably to cover their arse. However, legally in law as far as I can see you would not be breaking any rules if you carried out training, under 18 hrs per week, while in the USA on a holiday visa unless there is some INS place where it is specifically written that you require an M1 visa to fly. I have never seen this place. As far as 141 goes, there is no reason you cannot train under part 61 at a part 141 school, many people do it. Schools with visa approval must be part 141 schools but they can also conduct flight training under pt 61....but this probably goes back to this thorny visa issue, part 141 probably does require a visa as the training will be more than 18 hrs per week.
I have never held an M1 visa, though did hold a B1/B2 OCS visa when I did my flight training.