Does a card like Force Push (restriction box says Rebel Force User) also count for the command card Protect the Old Ways or are only cards with a pure Force User restriction box valid?

Force Push shows rebel affiliation icon and Force User trait , and Protect the Old Ways refers to the trait only.

Many Command cards have a restriction box listed above the ability. When playing a Command card that lists a trait, figure name, and/or affiliation icon, it must be played on a single friendly figure that matches all of the restrictions listed.