What you describe is skill not status (except for the leader of course). Skill is independent of status yes, but that's nothing new. Both are correlated of course. Skill can e.g. be used to gain status.

Indepenent of that I agree that status is/can be multidimensional esp. if you are in different groups.

You might explicitly point out that status is in the mind. Perhaps in the environment of evolutionary adaptedness, everyone's maps of the group and hierarchy converged, but now with n ~= 7 billion, they diverge wildly. It's not unlike how hearts, lungs, and brains ceasing to function were once concurrent events, and now they are not and we often feel confused. There have been situations where I thought I was low status and turned out to be high status, and vice versa. What qualities correlate with status differ between individuals, but we do all seem to have an internal Status-O-Meter that causes us to make the single-variable-dependence error in the first place. Figuring out how to affect that meter as generally as possible and figuring out whether or not such efforts are necessarily zero-sum does seem useful.

I agree with those who feel as though we're lumping a lot into one word. I like shminux's description below of status as influence. Maybe you could define a high-status person as a person who makes all of the local maps converge on a map with the influencer at a high place in the hierarchy.