Discussion below this answer involves my feeling that calling halo orbits in very asymmetric systems (e.g. Sun-Earth or Earth-Moon) as orbits around the primary, in 1:1 resonance with the secondary.
Thinking about this has led me to ask something specific:
Question: Is the Gateway's halo orbit bound to Earth? Would the Gateway remain in Earth orbit of the Moon disappeared but drift away from the Moon if the Earth disappeared?
For more on the Gateway's orbit, see