I'm not sure there's any consistent nomenclature. With eggs vs. without is sometimes called French Vanilla vs. just Vanilla. But you'll find plenty of Vanilla recipes with yolks just called Vanilla. And for other flavors, you'll find tons of recipes with yolks still called ice cream. You're technically correct but I'm not sure how widely used "frozen custard" is.

there is even a third option! besides ice cream with no eggs and ice cream with cooked eggs, there is ice cream with uncooked eggs. i often hear the latter two options as "custard" vs "straight method".