Interperting the AOPA guidlines - it reads that you must have an appropriate visa to train, however a paragraph later it also says that the AFSP system will reject an applicant without the appropriate visa.
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Visa notice: Taking flight training without an appropriate visa could be a violation of your immigration status and could result in your arrest and removal from the United States; therefore, it is important that you have a visa that permits you to take flight training in the United States. If you do not possess the correct visa, or if you have questions pertaining to your visa status or the appropriate visas for flight training, please contact your local Bureau of Citizenship and Immigration Services at 800/375-5283 or www.uscis.gov or the State Department Consular Affairs Office for assistance. The AFSP will deny flight training requests from candidates who are present in the United States illegally or who do not have an appropriate visa for flight training. Fees paid for denied applications are not refundable."
I have recently seen an approval via AFSP ( AKA TSA) for an Initial Instrument Rating that was approved on a B1/B2 tourist visa - - hmmmmmm that begs the opinion that in the eyes of the AFSP an unexpired tourist visa is sufficient (not visa waiver though I guess) to commence training.
In my humble opinion and from my research it appears that there is huge confusion (possibly deliberately contrived) out there relating to this. As I understand the situation presently there are guidelines pending that will require either an M1 / J1 or a long term visitor visa such as L1,E2, H1/2/3 , etc to train in the US but as of yet this interim rule has not been fully confirmed (this was mooted back in late 2004) and until then any unexpired visa will be sufficient to gain approval for an Initial rating such as Private and Instrument. As it stands currently you do not even require a TSA (AFSP) clearance for Commercial or Add On Ratings to existing FAA licences.