"the majority of" vs. "a majority of"

Question

Is there any difference between these 2 sentences? Do they mean exactly the same?

The majority of men does not have a moustache.

A majority of men do not have a moustache.

I am using a singular verb in the first example, and a plural verb in the second example, because this seems to be the rule, though maybe it is a wrong rule:

https://www.grammarphobia.com/blog/2007/09/majority-rules.html

P.S. The question is derived from the below sources, which provide inconsistent, unclear, hesitant, inconclusive answers. Let us try to solve it once and for all!

the majority of + plural noun + singular vs. plural verb

Most vs. Majority

What is the difference in meaning between "A majority of" and "The majority of"?

Answer 1

Whether it's "A majority" or "The majority" is irrelevant here. The determining factor is "a mustache".

A mustache cannot be shared, so the having must be related to the individual members of the majority.

The majority as a whole cannot have "a mustache". Each individual does or does not have a mustache.

So "The" or "A" majority of men do not have a mustache.
Or "The" or "A" majority of men do not have mustaches.

One would use the singular only when talking about the majority itself rather than the members of that majority.
"A majority of men has been elected to the senate." (as compared with a majority of women).