For some reason I can not figure out how to phrase this question in a Googleable way.

In Europe, the Schengen Area is a Common Travel Area. As a US citizen, I can spend 90 days at a time in France, then I need to leave for 90 days. And I can't just go back and forth from France to Italy because they are both in the same area for that 90 day clock. But, for example I can go from France to Ireland because Ireland is in the EU, but not in the Schengen area, so my time in Ireland wouldn't count against time in France.

Ireland and the UK also use the term "Common Travel Area," but everything I have read only applies to UK and Irish citizens, so they seem to be using the term completely differently than the Schengen rules. And some of what I have read pre-dates Brexit, so I can't quite figure out if any of it still applies. Lots of travel websites with great SEO make frustratingly vague statements about "not needing a visa for a short stay."

So, all of that said my question is actually pretty simple. Does the UK/IE "Common Travel Area" mean that they are a shared clock for a US visitor like France and Italy? Or can a person bounce back and forth between UK and IE, and is there some optimal strategy for that? And is there any subtle complication with the Schengen area if a person were doing a circuit between UK, Ireland, and France?