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.
Use of the infinitive to define verbs is a remnant of classical education. It originates from the conjugation of verbs in Latin. Because Latin uses inflections (or suffix-*ish* particles) to denote mood, person, number, tense and voice, there are a multitude of possible forms for any one verb, depending on how it is being used. Memorising the conjugations of verbs and the declensions of nouns is therefore essential to comprehension. Fortunately, the full conjugation of a verb can generally be derived from only its Principal Parts (hence this is how a verb appears in a Latin dictionary). These are the the present indicative first person singular, **the infinitive**, the perfect infinitive first person singular, and the perfect passive participle: amo (I love); amare (to love); amavi (I loved); amatus (loved). Of these Principal Parts, the infinitive is the least liable to variation in usage, and thus it is the most convenient and clear way to describe a verb.