When is a backgammon not a backgammon?
A backgammon occurs when you have borne off ALL your checkers, and BOTH of the following are true: 1) Your opponent has at least one checker in your home area and 2) Your opponent has borne off NO checkers.

In this situation, your opponent had ALREADY begun bearing off checkers when he got hit. So the fact that this checker was in your home board has no bearing on "backgammon." (And if he had gotten it out of your home board before you bore off, there would have been no gammon.)

The cube can operate with or without a gammon or backgammon, so you get the value of the cube in this instance. If you're playing "Jacoby Rule," you can't score a gammon or backgammon without the cube's being turned.