See SN 46.3
sati-sambojjhanga remembers Dhamma-teaching, then thinks (vitakka) about that Dharma.
Dhamma-vicaya-sambojjhnaga investigates that Dharma-teaching recollected by sati, by means of vicara (evaluation) paññāya (same word as the sampajano, what B. thanissaro translates as 'alertness').
ātāpi = ...
From Khuddaka Nikaya's The Questions of King Milinda (translation mine):
‘Venerable Nagasena, what is the characteristic mark of vitakka*?’
‘The characteristic mark of vitakka is "placing" (Appanā), Your Majesty.’
‘Give me an example.’
‘For example, Your Majesty, like a carpenter takes well-finished boards of wood and places (appeti) ...
Sati means to 'remember' (SN 48.10) or 'bring to & keep in mind' (MN 117).
Sampajana is situational wisdom; the right or thorough understanding of or for a situation.
Sampajana is from the words saṃ & pajāna (cp. pajānāti). It appears suttas such as MN 118, AN 3.121 & SN 12.28 indicate mere pajānāti might be a level of knowing or recognition ...
In line with what I said in my other answer, both are descriptions of self-awareness, of not getting carried away by the circumstances.
This does not refer to "zombie bare awareness" executed mechanically, but to normal, pragmatic, careful self-awareness of an intelligent person.
Mindfulness is not "awareness". Mindfulness is remembering, as follows:
When one has heard the Dhamma from such bhikkhus one dwells withdrawn
by way of two kinds of withdrawal—withdrawal of body and withdrawal of
mind. Dwelling thus withdrawn, one recollects (anussarati) that Dhamma and
thinks it over. Whenever, bhikkhus, a bhikkhu dwelling thus ...
And how, bhikkhus, does a bhikkhu exercise clear comprehension? Here, bhikkhus, for a bhikkhu feelings are understood as they arise,
understood as they remain present, understood as they pass away.
Thoughts are understood as they arise, understood as they remain
present, understood as they pass away. Perceptions are understood as
they arise, ...
Sati is often translated as "mindful of." So Ānāpānasati means: "mindfulness of in-and out-breathing" or "conscious of in-and-out breath" However, part of the act of doing this is partially somatic, your body experiencing itself, your emotions, thoughts, etc. experienced as they are in and of themselves. It's not a purely mental exercise. It's not reflecting ...