Question: 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)”.

Answer: 
I think there are a few factors here.

One is that, despite the use–mention distinction familiar to linguists and philosophers, most people don't *fully* distinguish uses from mentions; it's most comfortable to define a word by using it as the subject of a sentence, and with verbs that means using either the to-infinitive or the gerund: "To dance is to move your body artistically", "Dancing is moving your body artistically", "Dancing is when you move your body artistically", etc.

Another is that, due in large part to analogy with Latin, there's a grammatical tradition of treating the to-infinitive as if it were a single unit that should not be "split" (because its Latin counterpart is a single word: *esse* "to be", *habere* "to have", etc.).

A third is conventions resulting from the above. You point out that Merriam-Webster has an entry for *dance* rather than *to dance*; but its definitions all start with "to"! (Likewise, for the countable noun, the headword is *dance* rather than *a dance*, but the definitions all start with "a" or "an".) This distinction is a bit too fine for most people; granted, no one bats an eye at "The verb *dance* means 'to move one's body rhythmically'", but it's all too natural to make it consistent by dropping a "to" ("The verb *dance* means 'move one's body rhythmically'") or adding one ("The verb *to dance* means 'to move one's body rhythmically'"), the latter usage being the one that caught your attention.
So, the verb "to dance" is meant to describe the act of "dancing", which is also a verb. However, the verb "dance" is used to actively convey the action in the present. So, if you were to say that you wanted to "dance", you would then use the phrase "to dance" to help describe your desires. If you wanted to use the word "dance", you would use it when you wanted to actively carry out the action. 
It's also based on the situation you're using it for. That's the key difference.