What is the most “genuinely Buddhist” kind of mindfulness?
If you look at this article which describes The Noble Eightfold Path,


Chapter VI: Right Mindfulness (Samma Sati) talks about "bare attention"
Chapter VII: Right Concentration (Samma Samadhi) talks about "centering the mind on an object"


So I think they're both Buddhist, but one is called "mindfulness" and the other "concentration".