What will happen will depend on exactly what kind of space your path is embedded in and how the ribbon's movement is constrained by the path.
To simplify the answer,
I'll assume everything is embedded in $\mathbb R^3.$
If pulling on the ribbon forces the ribbon to follow the shortest path along the surface, but cannot lift the ribbon any significant distance off of the surface, then pulling the ribbon may tell you that your path is two-dimensionally curved (because then the ribbon gets pulled toward one of the railings).
In that case, however, the walkway's embedding in $\mathbb R^3$ might be a truncated cone rather than a perfectly circular right cylinder;
either of these shapes is topologically equivalent to a general cylinder and not to a Möbius strip.
On the other hand, if the Möbius strip is embedded in $\mathbb R^3$ in the same way as a long strip of paper whose ends you have taped neatly together, the shortest path from the post along the path to the post again is right down the middle of the path, so if all the ribbon does when you pull on it is move to the shortest path along the surface, it will act just like it would on a perfectly circular right cylinder.
But suppose that along any part of the walkway that is embedded in $\mathbb R^3$ in a way such that part of the path you walk on is curved "upward" in three dimensions, tightening the ribbon causes it to rise off the surface of the walkway up into space.
If the walkway is a perfectly circular right cylinder,
pulling on the ribbon will either tighten it around the middle of the path or lift it off the path until you have collected the compete loop of ribbon in a pile around the post, depending on whether the path is on the outside of the cylinder or on the inside.
More generally, a two-dimensional cylinder can be embedded in $\mathbb R^3$ in such a way that each side of the cylinder has both "upward" and "downward" curved sections, such that wherever you put the post (on either side of the surface) you can collect all the ribbon by pulling on it.
Every embedding of a Möbius strip in $\mathbb R^3$ will have an "upward curved" section, because at some point the surface has to bend somehow,
creating a convex side and a concave side of that section of the band, and you walk over both sides of that section.
If the ribbon can lift off the surface on the "upward curved" sections,
it seems inevitable that the "lifted off" parts of the ribbon will eventually pull other parts sideways off the walkway.
Eventually, you should end up collecting all but a small section of the ribbon, and that section will be wrapped around the walkway.
That is, you'll see it go over each of the railings on either side of the walkway near you.
But if we're in a world where the ribbon is allowed to go over the edge of the walkway and around to the other side, then why can't you go over the edge of the walkway and around to the other side?
If you can do that, it is possible to use some of the techniques from the answers to
the predecessor of this question,
such as making a line of paint along the walkway.
In fact, after laying down the ribbon from the post back to the post, you just need to look on the other side of the walkway to see whether the ribbon is on that side too, and then you'll be able to tell whether the walkway is a cylinder or a Möbius strip.
