I would claim that mathematics is the systematic exploration of idealization and human intuition. The objects studied are real only in an idealized sense, and the operations must obey idealized rules that approximate reality in narrow ways that minimize acceptance of external data.
So I would not claim that it is particularly about the integers, but your last statement fits my experience best.
The first situation is actual Platonism, the second is Formalism. These two approaches dominate the field in the sense that "Your average logician is a Platonist on weekdays and a Formalist on Sunday."
The third position is most clearly reflected by the project of Intuitionism, which tried to resolve the issues of Russel's paradox, etc., by questioning the natural force of negation and considering mathematics more a joint psychological endeavor that requires the investigation of our shared intuition, rather than a reflection of external or formal constructions.
Unfortunately, changing the meaning of mathematics requires reconstructing what is already known in another form, and such projects do not broadly capture the imagination of working mathematicians (though it makes better headway among those drawn to other computational disciplines.)