A voltage regulator allows the load to take the current it wants, while forcing the terminal voltage to remain constant.
A current regulator allows the load to determine the terminal voltage it wants, while forcing the current to remain constant.
Either can be implemented with a mainly voltage-output stage (like an emitter or source follower) or mainly current-output stage (like a collector or drain). The only difference is which of the output voltage or output current is sensed and fed-back to control the output stage.
To address your main point, voltage is not the 'cause of current'. At least, not in the sense that stops 'current is the cause of voltage' also being true at the same time. You set up the conditions, and current flows. While you can calculate numerically what's happening, arguing around the word 'cause' is going to confuse you.