Sorry av8tor, you are correct about the scope of the visas, albeit repeating commonly available info that has been done to death on here already, but there are a number of inaccuracies in some of the other things you say
You dont have to go to a part 141 school
Quite a number of flying training tasks require neither a visa nor tsa, nor a green card
A tourist visa IS quite acceptable for some types of training
See b2n2s post for activities acceptable without a visa, he is quite correct about those, some basic research or simply asking an immigration officer will tell you where the truth lies