Warning: This question is not purely mathematics, and requires a bit of "intuition" and "feel" for what I a beginner to topology, must be thinking, in order to be answered satisfactorily. Apologies, ahead of time.
In dabbling with topology I asked the following question:
Is it necessary to specify "how we glue something" in topology
Where it was concluded that the two different constructions of gluing points on a sphere I was considering, were the same (this naturally also follows in the lower dimensional case of circles in 2d).
But I realized, my problem is one of vocabulary.
Considering a similar thought experiment starting with a circle in 2D:
(Visualized here:) top half is the first construction in 2 steps, bottom is the other construction
https://awwapp.com/s/9541dcdf-c314-4832-9b3c-ad8415aa3b23/
Now the thing is that these two shapes "are different" in more than just an obvious geometric sense. There is no way to turn the bottom shape into the top, without crossing the boundary, if we constraint both to only be deformed in 2 dimensional ways, and if deformations are smooth (I mean this in the loose english sense, but coincidentally i might actually mean it in a more rigorous sense, see below), and if during these deformations, if we suspend time, at any point, and pick any point in the shape, we will be able to tell where that point "traces" to forwards in time (so multiple points can never map to the same place at same time, even in a limit)
Both shapes are then, homemomorphic, ambient-isotopic, but most generally they are NOT ____, and this is the term I want to find.
I have a feeling I might be referring to:
https://en.wikipedia.org/wiki/Diffeomorphism
But I worry that might be too specific, or perhaps there is a pathological diffeomorphism, I didn't consider earlier, that I want to disallow.
Actually you can prove that there is no diffeomorphism from $\mathbb{R}^2 \to \mathbb{R}^2$ that maps those two pictures to each other, because the second one has a point $z_0$ sucht that $\frac{1}{z-z_0}$ integrated along the second curve is $4\pi i$ but there is no such point in the first picture.
Since you're looking at something like branched knots I would say that ambient isotopy is the right equivalence relation. (For why isotopy is not the right equivalence relation just google "bachelors unknotting").