There's a very intuitive, visual explanation of how the equivalence principles relates to the bending of light here, which includes an explanation as to how the geometry of space enters the picture in terms of link up up infinitesimal patches of space together.
Note that it the freely-falling frame is important here because according to the principle, physics should follow special relativity (over a small enough patch). That's what allows one to conclude that the light beam should be straight in that frame... and therefore must bent in the other.
But answer your question directly, no, it is simply not the case that the equivalence principle is "the same thing as" the curvature of spacetime. Rather, the equivalence principle establishes that gravity and the geometry of spacetime are inextricably linked, but it does not by itself prescribe a particular geometry that spacetime must follow.
Einstein's version of the equivalence principle, from which the deflection of light follows, is:
The outcome of any local non-gravitational experiment in a freely falling laboratory is independent of the velocity of the laboratory and its location in spacetime.
In any spacetime geometry, a small enough patch looks flat, so in a free-falling frame spacetime looks just like special relativity (to first order). But this is true in any theory of gravity in which gravity manifests as spacetime geometry with gravitational free-fall following a geodesic. It does not have to be the geometry predicted by general relativity.