At some point the definitions become almost circular and opinions about what it fundamental have shifted a bit over the centuries. The cgs system of units -- which differs profoundly from SI in the treatment of electromagnetism -- was associated with those who viewed D and H rather than E and B the most fundamental. I'm quite happy with the level of theory used being appropriate to solve the problem at hand. There's always a bit of wiggle room around exactly what that problem is, however ;-)
It doesn't matter in terms of the math (in the vast majority of situations), so while the conventional idea of electric flow is incorrect, we keep it anyway.
Current moves far faster than electrons. it is more similar to a wave in the ocean with the electrons being the water molecule.
As a result, and counterintuitively for most, the speed of electrons will give you a completely wrong answer for when a light will turn on after you flip a switch.
Current is the movement of charges. It cannot “move” faster than said charges. (Or, perhaps, you meant the electomotive force that makes the electrons move along the wire, then sure, that thing spreads pretty quickly.)