I don’t believe the styles are any less ‘true’ because they display 4 beats to the bar. You could write music as notes with no bar dividers at all, and no time signature, and it would still be the same piece of music. But it might be hard for the musician to read, and would provide no information on how it should sound rhythmically.
Division into chunks and time signatures evolved to make it easier to read and write music on paper along with conventions surrounding them to give the musician additional information on how to play it. So for example if he sees 6/8 jig time written, he immediately has a feel for the rhythm.
But none of this matters when you don’t need to work it out from paper, or read the music, because you can hear the rhythm anyway in a style. As Fred says 2/4 and 4/4 are really just a bar length difference, the beat is the essentially same. 12/8 is actually closely related to 4/4 being four beats to the bar, just four sets of triplets, and is often actually written as 4/4 even on sheet music, where you might see something like 4/4 doo wop, rather than written as 12/8. In the same way 6/8 is related to 2/4, which in turn not too different from 4/4.
But it’s all fairly irrelevant for style management.
Mike