I don't know exactly where to draw the line on "the vast majority," but surely it must be higher than the bar for a simple majority, which is "more than half." If you want to describe something in the lead but under the 50% mark, the word you're looking for is "plurality."
Yes indeed, both meanings are possible in most contexts.
In US English, when speaking with the mathematical precision, majority means absolute majority (more than half) and plurality means relative majority (more than anyone else). British English does also have the term relative majority like in French, though I don’t know if this is used in mathematics.
But like most other dictionaries in both English and French (with some exceptions like l’Académie Française’s dictionary), Merriam-Webster tries to describe how language is actually used in the real world and not some theoretical idea of how it should be used.
Therefore, since “majority” is often used to mean either absolute or relative majority when speaking in a less precise context than mathematics, a general-purpose dictionary like this one lists both meanings. A mathematical dictionary from the US (again I don’t know about the British equivalent) would list just the absolute meaning.
As an Australian English and Indian English speaker and a mathematician, I have never heard the word plurality outside of discussions of the US political system.
I have seen nitpicking on whether the word majority is the right word for a relative majority, but only seen plurality offered as an alternative by American English speakers who are also students of the American political system.
I would almost never expect anyone to say "the plurality of cars sold are Toyotas", for example.
In French it's not the case, you can have relative or absolute majority, which might explain my confusion.
According to this definition https://www.merriam-webster.com/dictionary/majority : "c : the greater quantity or share" that also seems to be a possible meaning in English