In contrast with my esteemed colleague RichardKennaway, I think it's mostly #2. Before the Peano axioms, people talking about numbers might have been talking about any of a large class of things which discrete objects in the real world mostly model. It was hard to make progress in math past a certain level until someone pointed out axiomatically exactly which things-that-discrete-objects-in-the-real-world-mostly-model it would be most productive to talk about.

Concordantly, the situation of pre-axiom speakers is much like that of people from Scotland trying to talk to people from the American South and people from Boston, when none of them knows the rules of their grammar. Edit: Or, to be more precise, it's like two scots speakers as fluent as Kawoomba talking about whether a solitary, fallen tree made a "sound," without defining what they mean by sound.