Bill Kristol is actually part of the movement known rather derisively by real conservatives as “neo-conservative”, people who came late in life to the conservative movement, and think they can”improve” conservatism by making it into a statist, big-government party.
Definitely a branch of the Uniparty. Separate and apart from what are commonly known as “RINOs”, which are only thinly disguised moderate Democrats, and years back, would have been known as “blue-dog” Democrats.
I always knew what a NEOCON (new conservative) was, hadn’t heard the term PALEOCON (ancient conservative) until my daughter called me one.