What is the most “genuinely Buddhist” kind of mindfulness? I’d like to hear some reflections on mindfulness. I’m not talking here about meditation on the three marks or any kind of visualization, but mindfulness. 

I am quoting here from B. Allan Wallace’s The Attention Revolution. 


  [One] approach views mindfulness as nondiscriminating,
  moment-to-moment ‘bare awareness’ [and the other] characterizes
  mindfulness as bearing in mind the object of attention, the state of
  not forgetting, not being distracted, and not floating.


Which one of these 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".