It's entirely possible the claims were somehow automated or involved some drone in an office blindly filing DMCAs with links to the site (just a step above auto

It's entirely possible the claims were somehow automated or involved some drone in an office blindly filing DMCAs with links to the site (just a step above automated). A counter-claim is enough of a red flag that they'll stop to take notice of what exactly is going on and who they're filing against. If this is the case, they're more likely to drop it since pursuing Tweets isn't as effective as going after the actual site itself. Either way, if you do decide to fight it, you need some legal counseling unfortunately, hence the effort bit. If you prefix your Tweets with "Job Listing: " or just "Listing: ", that may deflect responsibility more in the future. Also on the Twitter description maybe an added disclaimer is warranted.