Mindfulness means remembering to practise Buddhist principles.
I know this is the Vipassana forum, but my strong personal feeling is that meditation is a neurological mental state achievable by many means. I think that neither Buddhism, nor any other religion, has a monopoly on how to enter that state. For instance Christian religion has a notion of centering prayer which is nearly word for word identical to Vipassana in terms of execution. And it's quite trivial generate 100% secular instructions for what is, essentially, Vipassana. And they work.
That said I respect Buddhism. I believe it is a treasure trove of information related to meditation. And its teachings and its adherents should be respected.
Although... I think I read a hallmark of Buddhism, maybe it was only Zen Buddhism, was rejecting the word of your teachers in favor of direction experience.
To that end, my direct experience is there is a strong correlation between what I would call mindfulness and my conscious awareness of my peripheral field of view. I have meditated for several years and I'm very aware when my mind is chattering and when it is still. And I find if I'm aware of my entire field of view that my mind is still. And if my mind is still (and my eyes are open) then I'm aware of my entire field of view. I've observed this for at least 6 months so it doesn't feel like just a "passing stage".
It's essentially the polar opposite of "tunnel vision" which I experience when my mental state is the opposite of midfulness, when I'm lost in thought thinking deeply about something completely unrelated to my sense data. For example thinking of something in past or the future.
Note that I can never
perceive tunnel vision. In that state I'm not aware of my lack of peripheral vision. What I notice is when I exit tunnel vision. It's like right before the movie when they pull back the curtains to reveal the screen is really much wider than it was during the commercials. At that moment my mind goes completely still and I'm fully present in my immediately surroundings, in the present moment.
At any rate, perhaps I'm off on a introspective tangent and my observations are bunk and not useful to others. I could imagine that being the case.