I don't drop in here often, but I went through this in 2006 for my
FAA IR (there are some TSA/Visa notes in that article; not necessarily up to date) and it's amazing that the same arguments go round and round 6 years later.
Do be careful with getting the I-20 and then changing schools.
When I started on the FAA IR in 2005 (the FAA PPL and CPL I did in the UK, at now defunct outfits) I was going to go to a US Part 141 school called CRM, who used a UK instructor as their "agent" here. They refused to communicate with me directly (as I found out too late). I went through the Visa process with them and they sent the I-20 to the instructor's UK address. No problem, but he was abroad for a while so the I-20 got lost.
I then cut my losses and did another I-20 with the school I eventually used (Chandler Air Service at KCHD). They were great and I completed the IR there no problem.
2 years later I flew to the USA for a 2-day private visit. The Immigration official found on his computa' the original CRM Visa, with a note (from CRM apparently) that I started training but did not complete. His little brain went Click Click Click (his last job was MacDonalds but this time he had a big gun) and he thought "this man trained to fly and not land". I got taken away and got a disgusting interrogation, in a room full of Africans who looked like they stowed inside some 747 landing gear, who were periodically yelled at by the guards, etc. I tried to talk to the 2 staff members present who were going through some paperwork but they were exMcD too (thick as a plank). 3 hours later I got out of there, accompanied by a Customs guard who did have a brain. I explained to him (as I tried with the others) that it would be trivial for them to check faa.gov and see that I had the IR, completed the training, etc. He may have been ex-McD too but hey he knew what "internet" was and he said they have no access to the internet
Luckily my connecting flight was 4 hours later so I didn't miss it.
After my return to the UK I tried to communicate with CRM to get this sorted. I was told by TSA (?) that only CRM can remove that "training incomplete" comment, but they refused to co-operate. TSA and Immigration refused to help.
Now I have a new passport without the M1 Visa stuck in it, so maybe I might risk entering the USA again
Unlike many Europeans, I am not at all anti-American (I do business with them all the time) but they do need a better system... and you need to be very careful changing schools if you have already got the M1 in the passport.