Other way around. Retrograde amnesia has consciousness, anterograde - inability to form memories going forward - may not.

I've suffered from a lot of anterograde amnesia from a couple forms; the most familiar to others would be from taking Ativan with alcohol, which more-or-less works like a roofie, general anaesthesia, or alcohol blackout.

In this form of amnesia, you still form short-term memories but they fail to consolidate to long-term when they should. As far as anyone but me can tell, I'm just tipsy and acting as normal.

So there is still a sort of memory forming, and I think we would assume that conscious-is-as-conscious-does, so if this definition excludes that, that would be a bit odd.