Satipatthana is part of constant practice, as satisampajanna.
Anapanasati is a particular way of doing satipatthana whereby one calms kaya- & citta-sankhara, as well as citta altogether, in order to facilitate a tranquil letting go, leading to seclusion from kamaguna and akusaladhamma & thereby enabling jhana.
Just as with satipatthana, the guide is the map, but the terrain for each of us requires skillful navigation; right view is the forerunner here because it is only with the cessation of doubt about wholesome & unwholesome that one becomes unerring in this. It needn't be the cessation of doubt such that a stream-entrant experiences, but that hindrance needs dealing with at least (as do the other four).
The piti of the second tetrad is the piti factor of awakening, arising due to energy (sammaviriya, as above) which springs from dhammavicaya, itself a result of sati. It can be present when doing the first tetrad, and is itself the condition for tranquility, which is the condition for samadhi (again, as above).
IMO, of course.