Let me introduce this question by noting that my training is in condensed matter physics, and my familiarity with gravity is superficial at best. Hence, this question may be somewhat naive.
In addition to the usual Einstein gravity that apparently governs our universe at long wavelengths, I have seen several different gravity theories discussed in literature. For example, I have heard of supergravity, topological gravity, bimetric gravity, W gravity, and maybe others. My question is: what makes a theory gravitational? It seems obvious to me that any theory that wants to call itself gravity better have a dynamical spacetime metric (correct me if I'm wrong). Is there any other ingredient that is essential to gravity?
Some corollary questions: Do all gravity theories have black holes? If so do they behave like black holes in Einstein gravity (e.g., obey the Bekenstein-Hawking formula)? Can all gravity theories be realized as string theories?