[auggierose]

As commonly defined, you are right, but if you have the ability to select among your models, then you can use that to sharpen your notion of categoricity by considering only a certain class of models instead of all models.

You can say for example, categorical with respect to a certain cardinality of the model, that is you are only considering models of that cardinality.

And just like that, if your semantic allows it, you can also say categorical with respect to standard models. But for that you of course need a clear definition of what "standard" means. First-order logic doesn't provide such a clear definition.