Use `client_artifact` for the logging target

We want to use the crate name as the target in the error / warning logs.

> new!
>
> ```suggestion
> target: snapshot.client_artifact(),
> ```

_Originally posted by @westarle in [#5148](https://github.com/googleapis/google-cloud-rust/pull/5148/changes#r2992092512)_

Once this is updated, we should update the docs to demonstrate how to filter logs by target (e.g. to get `DEBUG` logs for a specific service.)