Quote by PAllen
This was very interesting, but I didn't find any answer to my question my key question: how is it winding number of closed curve computed / defined against the definition of a differentiable manifold?

Yes, not a single word.
Quote by PAllen
If it can change while the manifold is considered identical, then it must be computed in a way that is not invariant.

I don't agree. If the Dehn twist is a global diffeomorphism (and if we agree that in 2 dimensions homeomorphic manifolds are diffeomorphic and vice versa  which does not hold in higher dimensions) then the two manifolds before and after the twist are identical  there is no way to distinguish them. Now suppose we cannot compute the winding numbers (m,n) but only their change under twists. Then this change is not a property of the manifold but of the diffeomorphism. So we don't need a way to compute the winding numbers from the manifold but a way to compute their change from the diffeomorphism (this is similar to large gauge transformations where the structure is encoded in the gauge group, not in the base manifold). I think for large diffeomorphisms there is some similar concept.