Why is this signature independent of the message?
Note that in the first variant given a signature and knowing the public key, the message $m$ is uniquely determined. Information-theoretically dependent means what you assume it means. An unbounded adversary who can compute all discrete logs, can thus easily figure out the unique $m$ in the first scheme.
In the second variant, the message is information-theoretically hidden (even for unbounded adversaries). Note that $m+zr$ is basically „an encryption“ with a secret key $r$ (and $r$ does not appear anywhere else). So for $v=m+zr$ you can always find a suitable $r‘$ for any possible choice of $m‘$. This information-theoretically hides the message.
The problem with your modification is as follows. In the second variant a message is $(m,r)$. If you modify it to $a^{x+xy(m+r)}$ then you can open your signature to any message $m‘$ by computing $r‘=(m+r)-m‘$ and providing $(m‘,r‘)$ as the new message. This clearly will verify. Thus, your modification does not yield an unforgeable signature scheme. For the second variant of the CL signatures, however, the scheme can be shown to be unforgeable as $z$ basically commits you to the $r$ value.