As far as i know, the Einstein equivalence principle (EEP) is the starting point to explain gravity as a geometric phenomenom. It allows you to link gravity with two geometrical objects: a metric, which defines distances, angles and the causal structure of the space-time; and the affine conection, which defines the pararell transport and therefore the curvature of the manifold. From this point, let's say you can follow 3 paths:
First of all, the Einstein one, you can impose the Levi-Civita conection which is totally determined by the metric. Thus, you only have one degree of freedom, the metric is the only field associated to gravity and the matter only couples to it, so particles follow geodesics of this metric. You can introduce aditional gravitational fields as long as they dont couple to matter and you still will have a metric theory.
Secondly, the opposite idea, try to explain gravity only with the conexion, so you get an affine theory.
Lastly, you can take the more general situation and supose that both objects are independent, getting a metric-affine theory. Both fields are associated with gravity and both can couple to matter. The latter has the inconvenient that partiles may not follow geodesics of the metric. To fix this you can allow only the metric to couple to matter, the so-called Palatini formalism.
Within this calssification, it seems that a theory in the Palatini formalism is a metric-affine one, but then i found that a metric theory has to satisfy this 'postulates':
Space-time is a differentiable n-dimensional manifold with a lorentzian metric.
Particles follow geodesics of that metric.
3.In local inertial frames, physics laws must be that same that in special relativity.
A theory in the Palatini formalism satisfies this points because you only allow the metric to couple to matter. The conection may be regarded just as an additional gravitational field that you can add to the theory, just like you can add an scalar field to get a scalar-tensor theory (Brans dicke for example).
Am I right? If so, that means that what makes a theory metric is the fact that only the metric is allowed to couple to matter?