C1/D is a crew member visa (flight crew / cruise ship crew etc..) and basically only enables one to embark or disembark in the US if memory serves me right.
But anyway, the reason Pt 141 schools are visa approved is because training under Pt 141 is a formal course structure which can take someone from zero to ATP. When training under 141 many of the requirements (like cross country / PIC time etc...) are dropped but are replaced by stage checks which the student is required to pass before continuing on to the next stage.
Pt 61 is informal tuition. Now really the M1 visa doesn't cover Pt 61 because....it is not nescessary....as it could well be a vocational course of under 18 hrs per week. And hence a 61 school can't be visa approved.
But the general consensus on Pprune is that you need an M1 (not that I have ever had one
) and so what the pt 61 schools do is go to their mate who runs a pt 141 school, and apply for the visa that way. They all do it by the way and I don't think you'll ever find a forigner training under pt 141...It has several distinct disadvantages for someone who say only wants a PPL or IR. Fine if you want to go zero to ATP.