If its the same as in the UK, Aerial Work occurs if the student pays for the aircraft i.e hire; it has nothing to do with the instructor being paid.
A number of schools operating in the US have applied similar interpretations but it does not mean it is correct. The reference in Stds Doc 39 was put there so that staff would be made aware that it was their responsibility to ensure they were instructing legally. Around 2000/2001 students were flying around Florida with no pilot licence at all, having trained for a JAA PPL, Night, IMC, and Multi and did not apply for anything, until they completed their hour building and returned to the UK with 150 hours; most of which was flow illegally in the US. The latter was of no concern to the UK CAA.