What's a Non-Nominative Subject in English?

The question is, can we really call these things subjects? Most of us are used to tidy agreement patterns. We tend to just do straight up SEO and the S is the subject on all sorts of verbal experiences. But in old-fashioned English or rather archaic English, we have me thinks. And that is an example of a non-nominative subject verb. The point is, think has been beaten down and turned into a nominative subject verb Whereas in the past, it used to be a dated subject verb.