They're both almost right...
In a metal (semiconductors can be weird) the electricity *is* the flow of the electrons. The electrons are sitting there (well, vibrating, or spinning around, or being quantum, or whatever model of the atom you feel like using) being Important Parts Of Atoms anyway even when there's nothing electrical going on, but if you squeeze an extra electron onto the atom one end of a conductor, the force shoves an existing electron over to the next atom along, and the one next to that, and so on until one pops out the other end. Like marbles in a tube, or people trying to cram onto benches the one on the other end pops out immediately, even though the one you just squeezed in will take ages to get to the other end. Because electrons all carry energy, energy is transferred to the other end of the conductor.
In an insulator, the electrons are all tied up with Holding The Atoms Together in such a way that they can't easily shove along to the next atom. Like a bench with arm-rests. So it doesn't conduct electricity. (Or, sometimes, only conducts electricity when you melt it.)
And yes, the electrons making your light work are moving at roughly snail speed (okay, they actually keep turning round and going back the other way 50 times a second so they don't actually get to go very far, but they're still moving at snail speed). Luckily the movement gets to the other end of the wire at the speed of light, so you don't have to wait half an hour for the room to stop being dark.
It's like when you've got a bike with a Chain Of Greater Than Average Length: Most of the time, the chain links are all just sitting there doing nothing, but when you pedal, one link pulls another, and the one after that and so on and the wheel turns immediately. But if you want to find the special link that lets you take the chain apart, you have to sit and turn the cranks several times before the one you're looking for pops out of the end of the tube (usually just when you get bored and stop paying attention, so you have to go all the way round again to get at it).
The magic word is "drift velocity", but that'll probably bring up some A-level physics.