Can you use reverse as regenerative braking in a 2 quadrant motor?
What is the difference between doing this and using regenerative braking (besides the obvious fact that you continue backwards after stopping)?

Regenerative breaking actually turns the motor into a generator instead of running backwards it puts a variable load (i.e. recharging the battery) on and &quot;re-generates&quot; power.

Does using reverse to brake the scooter as described above put energy back into the battery the same as regenerative braking would?

If you are actively driving current against the back EMF then no.  You are actually doing work by putting more power into the motor to increase the force in the opposite direction.

For drive manufacturers that sell drives with optional regenerative braking capability, what is really the difference between a 2 quadrant version of the drive? Can't the 2 quadrant version be used for regenerative braking with reverse?

You need a 4 quadrant motor to do regenerative breaking and be able to run in both directions.
The quadrants are like:
1 = Motor CW
2 = Regen CW
3 = Motor CCW
4 = Regen CCW
So there are either 1, 2 or 4 quadrant drives.
A drive that is 1Q is #1 only.
A drive that is 2Q can be #1 + #3, or it can be #1 + #2. Saying just &quot;2 quadrant&quot; is technically insufficient to differentiate.  You have a #1 + #3 from your description.
A 4Q drive is of course #1, #2, #3 and #4.  Which offers an active free quadrant for pulling power to regenerate and still have the ability to run both directions.
Take a look at this article on regenerative breaking.  And this article on drives.