Not so easy is it?
To my mind it makes sense that if the "highs" start at 2nd ledger C, then the "double-highs" would start AT the C above that. Therefore 4th ledger line G is "only" high G.
You can't arbitrarily start changing the "system" part way through an octave at different positions just because it makes it seem like you can play super high (high enough already!) By this argument you'd never change from "high F" to "Double high G". (Otherwise the D above "high C" could as easily be "Double high D" as it could be "High D"!
(Well, I guess you can.... but you'd be introducing a mess of confusion such as we see here...!!!!)
The bottom line seems to be that, as Senya has suggested, sticking to the "formal" system of Ci, Cii, Ciii (or C1, C2, C3) isn't as much "fun" but everyone knows what the heck everyone else is talking about!