By flat he means a "V" (aka linear)
edge bevel vs. a convex edge or
equal effective edge angle.
Seriously, though, I agree with stabman that a lot of folks
really think too much about this topic...but at the same time not enough. Either don't worry about it at all, or think about it a freaking TON until you realize all the time you wasted thinking about it.

Drink deeply from the well of knowledge, because
a little knowledge is a dangerous thing.
Or just save yourself the trouble and accept that a convex is thinner at equal effective edge angle vs. a linear edge bevel and leave it at that. Because the deeper you dig the more you'll realize that's how it is; I really don't want to make what can sound like such a conceited statement, but it's really as simple as that. I mean, you can actually
see it in the diagrams I posted.
Another way to think about it is in terms of fluid dynamics. Imagine a boat going through the water, with a certain given angle to the bow and width at the stern. Which will go through the water better? The convex. But which will have the greater internal volume? The one with a "V" geometry to the bow. The reason for the convex one going through the water easier (and the reason why basically every boat EVER is convex even though it's easier to build a V-shaped bow) is the same reason a convex has improved thinner geometry that smoothly passes through the water it displaces. However, if we could somehow lock the bow of that boat in a massive vice and try to snap it off, presuming we filled the whole boat with something like lead or concrete so it wasn't hollow,
the V-shaped one would be stronger because of the increased internal volume at the point of strain.