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?