No confusion here between visa and TSA. Just to clarify your use of the word 'any' though - Leaving aside the question of recurrency and type ratings, TSA is required for three specific items wherever in the world the training takes place: your initial FAA certificate (usually private is the initial but it can be cpl), multi and Instrument. Commercial does not require TSA unless it is initial faa licence. CFI does not require TSA. Class addons (eg sea) do not require TSA. Many other things such as complex or tailwheel endorsements do not require TSA.
Both your need for a visa to enter the USA and your best choice of visa depend on a combination of your personal circumstances and your choice of training method.