This is not generally true. Mechanics in 5e with a radius affect a sphere of that size, and for grids normally the rule is that grid spaces are affected if their center is within the radius. So TFA's token radius is probably meant for ongoing effects centered on the player - light cast by a torch, the range of a paladin aura, etc.

The funky diagonals you're describing are an optional variant rule that's often used to simplify movement, but I've never seen a group that measures spell effects with that rule.