What is Analytic Philosophy? What is Analytic Metaphysics?

This definition of "analytic philosophy", by Brian Leiter, seems to me to be ok:

"Analytic" philosophy today names a style of doing philosophy, not a philosophical program or a set of substantive views. Analytic philosophers, crudely speaking, aim for argumentative clarity and precision; draw freely on the tools of logic; and often identify, professionally and intellectually, more closely with the sciences and mathematics, than with the humanities. (It is fair to say that "clarity" is, regrettably, becoming less and less a distinguishing feature of "analytic" philosophy.) The foundational figures of this tradition are philosophers like Gottlob Frege, Bertrand Russell, the young Ludwig Wittgenstein and G.E. Moore; other canonical figures include Carnap, Quine, Davidson, Kripke, Rawls, Dummett, and Strawson.

The definition (preceding the examples) is methodological, rather than historical, and identifies three characteristic features:

1. Analytic philosophy draws on the tools of logic.
2. Analytic philosophy identifies with the sciences and mathematics.
3. Analytic philosophy aims for argumentative clarity and precision.

Admittedly these criteria will be rather vague in application to many individual cases. Still, it seems adequate to a good approximation. Also, no specific big metaphysical or epistemological doctrine follows (realism vs anti-realism; rationalism vs empiricism).
Now define "analytic metaphysics" by replacing "philosophy" by "metaphysics":

1. Analytic metaphysics draws on the tools of logic.
2. Analytic metaphysics identifies with the sciences and mathematics.
3. Analytic metaphysics aims for argumentative clarity and precision.

(It does not define what a "metaphysical claim" is, though.)