I'm not entirely sure you have your definitions straight, no offense intended.
In the strictest legal sense 'flight training' means something different then the every day common use of the word.
In legalize it's easy to be confusing in an effort to include all.
Since it's not clear to me if you made a typo or not: TSA/Visa is not required for the license validation aka the piggy back.
Again, no offense but I've been involved with M-1 training visas and their requirements for well over 10 years.
You wouldn't be the first one that fell though the cracks created by confusion and ignorance ( in the side of your immigration officer).