No reason to quote regulations as the
FAA has nothing to do or authority over visa requirements and background checks
The AOPA page you link to mentions you do not need the visa for
High Performance endorsement or
Complex or
Pressurized as mentioned in 61.31
From the AOPA page:
You are seeking flight training inside or outside the United States for U.S. airman certificate under 14 CFR. This rule applies to flight training that you could use toward a recreational, sport, or private pilot certificate; multiengine or instrument rating; or any initial U.S. airman certificate issued by FAA.
You do not need visa's or background checks for a host of things including proficiency training or flight reviews or IPC's.
Anything that leads to the application for a certificate or rating does.
