infinitives - Why include “to” when speaking about verbs?

I cannot grasp any semantic difference, in discussions of grammar matters, between quoting full infinitives (with “to”) or bare infinitives (without “to”). Thus I do not understand why the longer form is used at all.

In which circumstances is it better to speak of “the verb to dance” rather than “the verb dance”?

Where does the custom of using the full infinitive originate?

Dictionary entries for verbs do not seem to include “to”; for instance, in the Merriam-Webster and the Collins (current online editions), “dance” is an entry, but not “to dance”, nor “dance (to)”.