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I think there may be a little nuance that can be applied to the situation, where 2 different possible hats may be applied, and which both are valid views of type systems. View 1: Types are intrinsic In this view, it makes no sense to talk about a program/term independently of its type. In addition $\forall$s and $\exists$s are really "forall"s and ...
Cody's answer very nicely describes an important distinction. I would just like to point out a specific thing about the interpreation of $\forall$ as $\bigcap$. A typical way to get $\forall$ interpreted as an intersection is to use a category of PERs on a partial combinatory algebra. Let $\mathbb{A}$ be an untyped model of computation, such as the untyped $\... 2 It is unlikely to have a "name" because it is trivial: it can be solved with a hashtable, array, self-balanced binary search tree, or any other data structure that maps$x$to$X_i\$.