Sati, is mindfulness or at a stretch, memory.
Samadhi is concentration, focus.
Manasikara has to be something different from these two.
Silvant sutta: ‘A virtuous monk, Kotthita my friend, should attend in an appropriate way to the five clinging-aggregates as …a cancer, an arrow… alien’.
No meditation method based purely on focusing and/or mindfulness will make the meditator think an aggregate is ‘cancer’ - unless they reflected on it in that specific way. It is a verbal thought describing what he or she is observing. Now modern meditators may find thinking as part of their practice, a rather unusual thing. The Buddha said yonisomanasikara was used before each step in the Noble Eightfold Path, according to the suttas.
The Buddha did not leave verbal thoughts of topics of importance, out of the box of tools one could use to develop insight. In the Silavant sutta he states that doing so will lead to the fruit of stream entry and higher stages of attainment. Yonisomanasikara is also used for thinking about paticcasamuppada, one’s sila, and gives rise to Right view (the latter being reasoned concepts based on experience).
It makes sense that the untrained person would focus on a pleasant sensation and thinking many thoughts about it would give rise to craving and attachment (ayonisomanasikara). Mindful observance or pure focus is not what usually happens in such situations.
So manasikara means (verbal) contemplation- as it does in Sinhalese.
Yonisomanasikara means to contemplate deeper, rather than at the superficial level. Wise or deep contemplation might suit there. Ayonisomanasikara would be unwise contemplation as I see it.
With metta