If you're going to answer the question of whether a language has types, then you're already trying to see how it can be shoved into types. It's presupposed by the question itself.

You literally have to count the types by analyzing the grammar. So for the lambda calculus, you have lambda=1, halt. So does a single type mean no types or literally one type?

What's the advantage of thinking that 1 type actually equals 0 types? I don't see any, so in my mind, all languages are unityped or have a richer type structure. Whether a richer type structure is desirable is a separate question.