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.