Turing Machine where branches are resolved via arbitrary operator
Using any/all (a.k.a. or/and) gives rise to alternating Turing machines. Goldschlager and Parberry (On the construction of parallel computers from various bases of boolean functions, Theoretical Computer Science 48:43&ndash;58, 1986) consider the generalization to allowing arbitrary Boolean functions, and they call the resulting machines extended Turing machines.

To me, it would make sense to use the same term for what you're proposing. I suggest following the references in Goldschlager and Parberry and looking up who's cited them, to see if the term "extended Turing machine" stuck and if it's been applied to your scenario.