I hadn't heard about this case before, and so I don't know the details, but just from your own description, filing a preemptive lawsuit in response to an explicit copyright demand doesn't sound unreasonable - the advantages of doing so over waiting to be sued include venue selection and, perhaps more importantly, forcing the claimant to make their case, rather than letting it turn into a frozen conflict (with the possibility of them filing DMCA notices against app stores at any time). That doesn't imply they actually wanted to go to court.

Though I did just Google it, and throwing an extortion claim on top sounds... well, like a big mess.

They didn't need to do anything to "resolve their messy lawsuit" other than to withdraw the lawsuit.

That's exactly what they eventually did. I wouldn't have, if I thought a copyright claim against me were truly bogus. Maybe they decided it wasn't bogus.

Oh, sorry, I misunderstood.