Well, this is getting fairly interesting.

Barry, you said, "Your reference to the verb 'to hit' (I would say the verb 'hit') is also confusing. Your example 'The hit ball flew over the fence' is somewhat contrived. Once again 'hit' is not being used as a verb. The fact that 'hit' is capable of being used as a verb in a different context is of no relevance. You are actually using 'hit' as an adjective. Further to your previous advice to look up in in a dictionary I have checked the main dictionaries for the use of 'hit' in an adjectival way. I have been unable to find any such uses. Hence my view that your use of 'hit' is contrived."

Would it help to say that the "hard hit ball flew over the fence"? Perhaps I am using baseball instead of cricket.

As far as "ajar" and "to" are concerned. In syntactic terms they could be called equal to "shut". They modify.

But so much of this discussion and apparent confusion comes from mixing terms from morphology and syntax. These are different subjects with different lexica.

How a word is operating within a sentence is a matter of syntax. There the words "verb" and "adjective" can be confusing. It would be better to speak of "simple predicates" (or "gerunds" or "participial phrases" or "infinitive phrases") and "modifiers".
These latter terms will not be used by a dictionary, which limits itself to morphology, although it may give some contexts to show how a given word can be used as more than one part of speech.

Regarding my naming the verb "to shut" as "to shut", in my experience that is what linguists, careful grammarians, and careful teachers of foreign languages do. They name verbs by their infinitives. If you simple refer to the verb "to shut" as "shut", it could be the present, past, or past participle that you are talking about (or the latter being used as a modifier -- adjective, if you will).

"Am, are, is, was, were, be, being, and been" are all forms of the verb "to be".
"Go, goes, going, went, and gone". These are forms of the verb "to go".

I have very little idea of how this is all taught in the British schools, but here in America, before the general demise of grammatical knowledge, beginning around the 1970's, all of this, within our system, was quite clear.

I realize that this is getting long. But if you REALLY care about my approach to this, you should look at the videos on the Youtube channel, mrbisse1, beginning with video 42.1 and then stay with the ".1" series.

Frank