Why is the verb 'realise' used as a state verb and much less commonly as an action verb in English? @JJJJS
Traditional grammar doesn't deal very well with this and stuffs 'to realize' in as a 'state' or 'stative' verb along with 'to be' etc. There's supposed to be a grammatical commonality in terms of the non-use of the present continuous among state verbs, but there are loads of exceptions to that and in terms of the complement, realize takes an object either usually in the form of a clause (Bitter's 1st example) or a simple direct object (2nd example). Verbs like "to be" and "to have", truer state verbs, don't. It's a bit of mess, basically.
So, I recommend looking at a functional grammar (see diagram below) to get this type of thing clear in your mind. It's in general a much better tool for understanding how the specific language you're looking at works than classical grammar is as the latter is stuffed with illogical rules, some of which are borrowed from other languages (Latin, for example, in the case of English).
Anyhow, "realize" does represent a kind of state, a psychological (but not a relational) one, in the first example, the state of realizing something. So, it appears somewhere between pure relations and pure actions. The diagram below helps visualize the gradual transformation. Verbs like "to be" and "to have" are true state (or relational) verbs—they describe the subject either in terms of identification (e.g. "I am an animal trainer") or attribution (e.g. "I am scared" (intensive)), (I have a knife (possessive)), (I am in the cage (circumstantial)). Verbs like "to do", "to put" etc., on the other hand, are pure actions. Verbs like realize then in the first sense describe conscious thoughts, feelings and senses and behave in the grammar somewhere between pure "state" and pure "action" verbs hence why it's confusing to think of them as either and why they don't behave grammatically as either. (Though realize in the second sense of "made something happen" is, of course, much closer to a pure action verb).
(BTW The "Existential" category refers to phrases like "There is a problem", "There seems to be a hurricane coming" etc...)