If the current is increased, is there more charge flowing or is it moving quicker?

Problem

Current is the amount of charge that is flowing through a component per unit of time.

For a given voltage, Ohm's law tells us that if we increase the resistance, then the current must decrease.

But what's actually happening to decrease the current?

My reasoning so far - is it correct?

More resistance (if we're talking about something of the same size for simplicity) is more 'stuff' in the way (higher resistivity), so more collisions. More collisions means it takes longer for the charge to 'get through' the component. The charge is moving slower and so the current is lower.

Or, is it that the speed is always the same and that somehow if we have more resistance then it means there is just less charge able to flow, hence a lower current. If so, why?