You would weigh a different amount on the moon, but that is completely independent of how we label our units of measurement.

The pound a unit of force. It is also a unit of mass. Both units share the same name.

It's not independent because if I'm 80kg here, I'm still 80kg on the moon. Because gravity does not affect mass.

If I weigh 160lb on earth I'm told that I would weigh 26lb on the moon.

That makes sense if lb measures force because gravity affects force.

In short, this commonly shared "fact" is consistent with pounds being a unit of force, not of mass.

Or, I suppose, of lb at least sometimes being a unit of force.