What are all the types of topologically-relevant band degeneracies in contemporary 2D condensed matter research? What about those often ignored?

Cross-posted the question here: https://mattermodeling.stackexchange.com/q/4877/1766

In the study of 2D condensed matter systems, I have seen several kinds of band degeneracies. I call 'bands' the eigenvalues to the time-independent Schrodinger equation, solved over 2D k-space. By 'degenerate', I refer to when the eigenvalues to one eigenvector are the same as the eigenvalues of a different eigenvector at the same k-space point.

I have seen bands that are:

1. degenerate at certain high symmetry points (such as graphene);
2. 'fully gapped' (with no degeneracies whatsoever);
3. completely degenerate (such as the 4-band Kane-Mele model, where each pair of bands is degenerate everywhere over k-space);
4. degenerate at non-high-symmetry points (degenerate in such ways that make them 'accidental degeneracies'?)
5. degenerate at certain high symmetry areas (similar to 4., but also includes lines and patches of k-space)

My question is: which of the above are relevant to topological effects (or coincide with those)? I have seen 1-3 above pretty often, but am having trouble understanding 4 (and 5). On top of that, are there more kinds of degeneracies not listed above?

Consider the attached picture, which is a random band diagram for $$MoS_2$$ that I found online. To my understanding, people do not care about degenercies of the kind I have circled in red (i.e. of type 4 in the list above). This is because they tend to care about the physics around the 'valleys' indicated by red arrows (i.e. fully gapped, no degeneracies). They usually do this by working with an effective Hamiltonian focusing on a subspace of the complete Hilbert space. Then, for topological reasons, they might treat the entire 'bundle' of wavefunctions above and below the Fermi level (indicated by red arrows), regardless of the degeneracies circled in red.

However, it seems suspicious to me that the 'degeneracies' circled are assumed to not play a role in topological properties. To paint a naive/vague picture, consider Haldane's interpretation of topologically significant degeneracies of a 2-level model as Dirac strings/wormholes that connect two otherwise-separate genus-1 tori into a genus-3 torus (second picture, taken from these slides). So, I am looking for consolation/clarification on why we can simply ignore degeneracies that occur outside the regions of interest, in general topological concerns. Or if anyone sees my overall confusion, I would appreciate any guiding remarks.

Note: I am more concerned about the type of 2D topology that deals with Berry-like quantities (Berry phase, curvature, Chern / valley Chern, etc).