This is not a claim about a "message space of size 2". The message space can be as large as you want, and the second characterization simply says that, for every $m,m'$ you pick from this message space, and for every possible ciphertext $c$, the probability that $m$ is encrypted as $c$ is the same as that of $m'$ being encrypted as $c$, which is written as $\Pr[\mathsf{Enc}_k(m) = c] = \Pr[\mathsf{Enc}_k(m') = c]$.
Now, in regards to the sketch of the solution you give, it's not really true that it is "assuming" a message space of size two. You want to prove a claim about two fixed messages $m$ and $m'$ (namely, you want to prove that $\Pr[\mathsf{Enc}_k(m) = c] = \Pr[\mathsf{Enc}_k(m') = c]$), and you want to do it while making use of perfect secrecy, which states that, for every message $\mu$ and every ciphertext $\gamma$,$^*$ and very importantly, for every distribution $M$ over the message space, it holds that $\Pr[M=\mu|C=\gamma] = \Pr[M=\mu]$.
Given that perfect secrecy holds for any distribution, we can arbitrarily choose any distribution that helps us towards proving our claim. The solution you're proposing simply takes the probability distribution that samples $m$ with probability $1/2$, $m'$ also with probability $1/2$, and all the other messages are sampled with probability $0$. One can also say that the message space is "restricted" to $\{m,m'\}$, but what is actually happening is what I said just before. Now that we have fixed the probability distribution, we also fix $\mu = m$ and $\gamma = c$ first, apply perfect secrecy, then fix $\mu = m'$ and apply perfect secrecy again, to obtain different expressions that can be manipulated to obtain what we need.
In short, this is just an artifact of the proof since the claim you want to prove is only concerned with a fixed pair of messages $m,m'$, so you can restrict a probability distribution to only these two elements, and apply perfect secrecy to this distribution.
$^*$ Notice I use other names instead of $m$ and $c$, since the latter are fixed already in our context.
