>including complaints by people other than the website/domain owner:

Yes, they are legally required to listen to the copyright owner's request. In your linked example, that was the church of scientology. In your earlier post, that would be whoever was writing the tweets. Twitter could also choose to block their requests as the wayback machine would also grab their copyrighted code, but choose not to.

You're making a documented legal requirement sound like a shady partisanship. Unless there are examples of these people wanting their content archived and the IA refusing, I don't see your point. Anybody is free to tell the IA not to archive their copyright material.