I understand that Emma's is clearly determining class. It's telling us which class. So it makes good sense to call it a determiner.
I'm sympathetic to the idea that determiners are themselves a subclass of modifiers. That is, that all determiners can be considered to be 'modifying' their nouns. The way I'm using modify here is quite general—to mean something like 'giving definition to (or, in other words 'restricting the meaning of') the concept'. Put very simply, this means that a modifier tells us something about a concept, whether that concept be a noun, adjective or whatever. Modifiers give definition—a clearer picture to imagine, rather than a mere vague concept. Lewis Hamilton's bright red sports car creates a clearer mental picture than just car.
But as I said, I'm not a grammarian. You should probably go with what the majority of contemporary grammarians consider standard. I'd suggest reading Huddleston and Pullum for this.

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