Definition of perfect secrecy using ciphertext
I think they are similar but not equivalent. 

The first condition says that for any ciphertext $c$ all messages $m$ have equal probability to encrypt to $c$. The second says that for any message $m$ all ciphertexts $c$ have equally probability of encrypting $m$. 

So I think you can demonstrate non-equivalence as follows: 

Imagine a (somewhat contrived) scheme satisfying the first condition but not the second: The scheme has two disjointed sets of ciphertexts $C_0$ and $C_1$. Concretely, consider one-time-pad where keys and ciphertext contain one extra bit, simply indicating if we are in $C_0$ or $C_1$. Now, assume we pick the extra bit to be 0 with probability 1/3 and 1 with probability 2/3. 

Now if $c \in C_0$ and $c' \in C_1$ it is clearly not the case that 
$$
Pr_{k \leftarrow Gen}[Enc_k(m) = c] = Pr_{k \leftarrow Gen}[Enc_k(m) = c']
$$
because the probability that $Enc_k(m) \in C_1$ is twice as big as $Enc_k(m) \in C_0$.

Alternatively, I think you can show that condition two implies condition one. I.e. condition one is more general than condition two.