What flush stair nose means
There is no lip and no step at the front of the tread. The surface runs out to the nosing and stops there, so the top of the step is one uninterrupted plane from back to front.
It is also the quieter of the two profiles visually. Without a raised edge the step reads as one piece of material, which suits a staircase meant to continue a floor, not to announce itself.
Fixed floors only
A floor that is glued or nailed is held at its edges, so there is no gap at the front of the step waiting to be covered. The nosing can meet the floor directly and the joint between them can be tight.
A floating floor is the opposite case, and the stair nose page sets out why. In short, it needs room to expand at the edge of the step, and a flush profile has nowhere to give it.
The joint at the nosing
With no lip covering anything, the line where the floor meets the nosing shows on every step. On a well cut staircase that line is what makes the work look considered.
There is no lip to hide behind, so the fit at that line is what the eye reads on every step. A staircase built this way is a piece of joinery as much as a floor, and agreeing that with whoever is doing the work is part of choosing the profile.
Floors that take a flush nosing
Solid and engineered wood fixed down, glue down vinyl plank, and anything bonded to a subfloor instead of clicked together. All of those are held at the edges, and that is the condition this profile assumes.
It also turns up on staircases finished in the same material as a fixed floor running into them, since the point of the profile is that the two read as one surface across the top of the step.
Stair risers
The vertical face below takes a riser, usually from the same range so the two meet cleanly under the nosing.
Together they give the version of a staircase where each step reads as one solid piece, which is the look the profile exists for.