"Mindfulness of a thing" means samatha. "Just mindfulness" means vipassana. Sometimes they combine both and call the whole program "vipassana"...
Messy and inconsistent.
I like the terms "concentration" and "growing".
For me phrases like "gently staying with" or "quietly attending to" or even "remembering" (which is sort of what sati means anyway) work better than "concentration", but it seems like there isn't a perfect piece of terminology to describe the kind of effort/effortlessness balance that has to be achieved for samatha to work. But I feel that "letting go of distractions" is a better way around it than "concentrating one-pointedly" which tends to sound very effortful.
Mindfulness/vipassana/insight, I have no idea what a good phrase is. There are so many modifications of insight techniques that it's hard to find a common core sometimes.
That said, I believe the early Buddhist texts (i.e. the suttas) don't really make the samatha/vipassana distinction much if at all, this is a (relatively) recent development. And it seems like practicing one necessarily invokes some of the other (apart from during very refined states of samatha where there's no "bandwidth" for the kind of analysis and noting that happens in most insight techniques). I've often thought of vipassana practice as "applying the principles/mechanics of samatha to (xyz mental noting technique)", which might be why some traditions tend to have you do some form of samatha first before vipassana - because the mind needs a little training in being steady and dropping distractions before it can apply a vipassana technique in a (roughly) one-pointed way.
When I do the second one, I watch sensations come and go while refraining from moving my awareness in reaction to what I observe.
Is that what you do?
If we look at the suttas, the Satipatthana Sutta, or Discourse on the Establishment of Mindfulness, offers a recipe to remain mindful of the sense aggregates to achieve insight:
"In this way he remains focused internally on the body in & of itself, or externally on the body in & of itself, or both internally & externally on the body in & of itself. Or he remains focused on the phenomenon of origination with regard to the body, on the phenomenon of passing away with regard to the body, or on the phenomenon of origination & passing away with regard to the body. Or his mindfulness that 'There is a body' is maintained to the extent of knowledge & remembrance. And he remains independent, unsustained by (not clinging to) anything in the world. This is how a monk remains focused on the body in & of itself."