Comment by calf

That carryover is because Turing machines and Lambda calculi are equivalent models of computation, and not because "busy beaver" is some special concept. It's a function whose definition is about computational phenomena, so there's a version of it for any Turing-equivalent concept (like lambda, or Post Correspondence, or the Recursive Functions)

So there should be no confusion there, the "big four models of computation" are all equivalent and we first encounter/learn this in undergrad TCS class.