My question is that in this game, it seems as if the adversary is choosing the two messages. Isn't this not simply an eavesdropping adversary, but rather one that can choose the messages being encrypted?
Yes, the adversary gets to choose the two plaintexts. There are a few ways to think about why this is appropriate.
It is very realistic to assume that the adversary has a-priori partial information (imagine an encrypted protocol session where we know that all protocols start with a "hello" message, like SMTP) or even partial control (imagine a website where you type a username and the website responds with a message "Hello, <username>") over what plaintext is getting encrypted.
In order to cover these realistic cases, it is necessary for the definition to allow the adversary to influence the choice of plaintexts. Once you include that into your definition, you end up converging on the standard chosen-plaintext-attack-style definition, since it is the most pessimistic and conceptually simplest way.
If the security definition didn't allow the adversary to influence the choice of plaintext, then every time you used an encryption scheme, you would have to think to yourself: am I absolutely sure that no external entity had any influence over what I'm encrypting? The standard definition gives you a guarantee that you can more-or-less encrypt whatever you want (there are weird edge cases like encrypting the key itself, etc).
Note that quantifying over all possible $m_0, m_1$ is equivalent to giving the adversary choice of $m_0, m_1$. This is because for every $m_0, m_1$, you can consider the adversary whose only strategy is to choose those $m_0, m_1$.
Having the game itself choose the two message $m_0,m_1$ is not equivalent. Such a definition would not rule out the existence of "bad" plaintexts. Consider a scheme that says: "to encrypt $m$, if $m$ is all zeroes then reveal the secret key, otherwise do a reasonable thing." This scheme would be secure under this weaker definition (since all-zeroes $m$ is rare, assuming the message space is large enough), but not the standard definition.
One reason to favor the "adversary chooses" style of definition is that we are already quantifying over all adversaries. We don't need another quantifier.
Actually in other settings (e.g., public-key encryption, poly-time adversaries), the "forall $m_0, m_1$" style is actually stronger! I think this is contrary to your suggestion. Imagine the public key contains $f(x)$ where $f$ is a one-way function. The encryption scheme says "if $m=x$ then give out the secret key, otherwise behave as normal." Quantifying over all $m_0, m_1$ would include quantifying over $m_0 = x$, and the scheme would be deemed insecure. But no poly-time adversary could actually choose $m_0 = x$ (that implies inverting the one-way function), so the modified scheme is equivalent to the original scheme under the "adversary chooses" style, so it would be considered secure.