As shown by Holevo, Shirokov, and Werner in (https://arxiv.org/abs/quant-ph/0504204), there exist separable but not countably decomposable states (in fact such states are dense as they conjecture). Since every separable state on the tensor product of finite dimensional Hilbert spaces is finitely decomposable, your condition 1 does not imply 2. 

See the MO post 
https://mathoverflow.net/questions/332478/is-the-set-of-separable-quantum-states-closed/
for some relevant discussion (and my goof).