Replacing _init_coordinates.

Split out from #31894, for #32102.

Depends on #32009

CC: @egourgoulhon @vbraun

Component: manifolds

Author: Matthias Koeppe

Branch/Commit: u/mkoeppe/chart__parse_coordinates @ 3d1c44d

Reviewer: Eric Gourgoulhon

Issue created by migration from https://trac.sagemath.org/ticket/32116