the conflict between the "machine model"  ...

I'm not sure I can agree. The original post makes the point that you can see recursion in multiple ways, including machine-oriented (e.g. using a stack and stack pointer) and abstract, e.g. using an axiomatisation of self-reference, for example using the Y combinator which satisfies YM = M(YM).

Incidentally, stacks can also be given an axiomatic treatment indepented of concrete machine models.