Your expression (1) can be taken as the statement of the correspondence, this is called the "extrapolate" dictionary. Namely if I have some bulk field $\phi$, then the CFT operator dual to $\phi$ can be obtained by pulling $\phi$ to the boundary of AdS and stripping off suitable factors of the radial coordinate. In your case the bulk field is the metric and the dual operator is the stress tensor. To see that the dual operator is actually the stress tensor, note that the mass of the graviton is protected by gauge invariance so the dual operator with spin 2 should have dimension exactly equal to d. Generically the only such operator is the stress tensor.

See Harlow and Stanford for the proof of the equivalence to the other form of the dictionary, http://arxiv.org/abs/1104.2621.