I want to know if there is a standard terminology for this among set theorists working with element-based set theories like ZFC. I will follow the convention that a class in any given set theory is a definable class--it is all the sets that have some property $\phi(x)$ expressible in the theory. So it is generally not a legitimate entity in the theory, but we can talk about it in the theory.
Then I call a class a proper class if it is not coextensive with any set. In ZFC that is well expressed by saying a proper class is too big to be a set. But that already fails in Zermelo set theory. Zermelo set theory has a definable class of all finitely iterated power sets of the natural numbers, which is not provably a set but is provably countable. In Zermelo set theory we can still say a proper class is a class not contained in any set (since the axiom scheme of separation says the intersection of any class with a set is a set).
The matter is yet more complex in Bounded Zermelo set theory, where the separation axiom scheme only holds for defining conditions wilh all quantifiers bounded. In this theory a definable subclass of a set need not be a set. For example Bounded Zermelo set theory can define the class of all natural numbers $n$ such that there exists an $n$-th iterated power set of the natural numbers. But it cannot prove this class is a set. (See: A.R.D. Mathias, The Strength of Mac Lane Set Theory, Annals of Pure and Applied Logic 110 (2001) 107–234 doi:10.1016/S0168-0072(00)00031-2)
So definable class in Bounded Zermelo can be too complex to be a set (in the sense of quantifier complexity) even when it is contained in some set. But do people in the field use some other term than "too complex"?