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) ...
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When one has heard the Dhamma from such bhikkhus one dwells withdrawn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 Dwelling thus withdrawn, one recollects (anussarati) that Dhamma and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 ...