Update AutoFactory integration tests and docs for JDK 23 `-proc` changes

[cpovirk]

IIUC in #1962, we're just throwing the AutoFactory processor on the classpath and relying on javac to discover it there.

Our docs advise the same: https://github.com/google/auto/tree/main/factory#download

We should instead use annotationProcessorPaths, as we've documented for AutoValue for a while, as recently updated in #1957. (Also: AutoFactory has provided a separate annotations artifact for a long time now, so we can change the classpath to use that.)

I may make an attempt at this, depending on how the rest of #1962 goes and how hard it looks.

[cpovirk]

Oops, I am mixing up AutoFactory and AutoService. AutoFactory does not yet support the split (#1814). So there is nothing to do with the docs and tests until that split happens.

[cpovirk]

Well, OK, that's not true. We could (and probably should) still recommend putting auto-factory in annotationProcessorPaths, even though you'll also still need to put it in dependencies. And we can do the same in our tests.