[wtallis]

> "an activity using the material"

You cut off a few words that probably carry real meaning:

> "does not have actual knowledge that the material or an activity using the material on the system or network is infringing;"

So there's a (somewhat ambiguous) restriction that the only activities this clause is concerned with are those happening on the system or network operated by the service provider that wants safe harbor protection against liability for their users' infringement.

And to determine what activities can qualify as copyright infringement, you have to look in section 501 and the other sections it references; section 512 does not alter the definition of copyright infringement, it just adds nuance to who can be held liable for infringement.