They wish to receive flight training from an FAA-certificated facility, provider, or instructor that could lead to an FAA rating whether in the U.S. or abroad
IMHO that is meaningless. Take a dual-rated instructor (FAA CFII & JAA IRI, say) doing a training flight in a G-reg plane. No way is a TSA approval needed; there are many such instructors working every day, in the JAA scene. Nobody can control whether one of those flights ends up being used towards an FAA IR one day.
Before any FAA Examiner/DPE will conduct the Test, he/she will want to see TSA approval had been granted.
Not, IME, ever, on flights done in the variously distant past. He might well enquire regarding the 3 hours you did in the 60 days before the checkride... and I am referring to US based FAA examiners here only.