The map that starts this threat (the Gleason map) is the Polar Azimuthal Equidistant (PAE) projection, the same as the map on the UN flag altho the UN flag has the earth turned so Africa, instead of America, is most prominent. The map itself uses the word 'projection' - which necessarily means it is an attempt to express a round earth on a flat page. There are more than a hundred different map projections, all of them show some distortions when mapping an area as big as the entire earth. The Gleason map is not the only choice; in fact the Gleason map was done in 1892, before the first Wright Brothers' flight, so it lacks some geographic precision, and there are newer versions of the same projection.
But the Gleason map rests on the notion that Antarctica is spread for more than a million miles, all along the rim of the flat planet. And people who have been there (and hardly anyone was in 1892) can tell us that Antarctica is nothing like that.
There will not be a 'definitive' Flat Earth map until someone - anyone - locates, for real, with tangible evidence, the EDGE, the boundary, the limit, - y'know, the line where the known geography of the Flat Planet ends. It ought to be in every direction, if you go far enough. Most people expect an actual edge, a corner, a side going downward; although some theorize that there is uncharted territory that simply stretches onward beyond the limits of known geography, perhaps into infinity (which raises a question of where the sun, moon, and stars go when they sink below the horizon). Either way, until that boundary is found with some precision, we will not have a definitive Flat Earth map.
But once we have a definitive Flat Earth map, then mapping will be easy. No 'projections' with their complex mathematics. Mapping a Flat Earth on an equally flat paper amounts to a floor plan of the flat planet - the only math is a simple scale, the ratio of (e.g.) one inch equals ten thousand miles. Every map of the Flat Earth would then be a bigger or smaller version of the very same map.