What seems to matter is how the 'Law' is being interpreted by the Department of Homeland Security and the departments within it. For instance, it seems to me the recent security law that Congress rushed through in haste didn't work well in practice, and the money we pay to AOPA to represent us properly in Washington (ie payment of dollars to our politicians represents us better than voting for them) made the TSA re-intepret the law (after several attempts) in favour of US citizens.
Other laws, eg whether a non-resident alien needs an M1 visa to train for a PPL, are now being interpreted in that you do need one. I agree with englishal that you never used to need an M1, and nothing has changed in immmigration Law that now means you have to have an M1, so you shouldn't need an M1 now either.
So tell that to the immigration officer and expect to get put on the next plane back to the UK and banned from the US for five years.
englishal, don't bother with the green card unless you need to work in the US, you don't need the IRS hassle.