Conditional 2 can't be restricted to only "counter-factual present" and "unlikely(hypothetical) future", it also includes timeless,
-> I strongly agree with this argumentation.
Try to accept that that all conditional sentences are in a way hypohetical. The possibility of the realisation of the hypothetical situation ranges from effectively 100% ( the so-called 'zero conditional') to effectively 0% (what some of us call counter-factual). Context, co-text and the shared knowledge of speaker and listener usually make the intended meaning clear. If it does not, questions may be asked and/or further information given:
A: John arrived an hour ago
B: Well, if he arrived an hour ago, we must set off now.
C: If he had arrived an hour ago, he would be here ago.
A. He arrived! he called me half an hour ago.
B is almost certainly indicating acceptance of the factuality of A's first statement. However, intonation, particularly a heavily stressed 'if,' would suggest doubt about this. We cannot say absolutely..
C is almost cartainly indicating strong doubt, perhaps even disbelief about the arrival.
So in some cases, even 2nd conditional can be interpreted as unlikely timeless, a vague area between counterfactual and 1st conditional, but I guess it's not so common. I hope my understanding is following your explanation.
In most of the threads in which we have discussed conditionals, the people responding to your questions have stressed the importance of context. In this thread, Tdol mentioned context as far back as the second post in this threas.
But, let's face it, this is true of most utterances in English. A 'simple' utterance such as , "I love you" can have an extremely wide range of shades of meaning, including, depending on the context, "I hate you" and "You love me".
The surprising utterance "Stalin and Hitler were devoted to each other all their lives" becomes less surprising if we realise that the speaker is talking about two dogs given 'amusing' names by their owner.
I have chosen admittedly far-fetched examples in the second and third paragraphs above, but they perhaps help make the point that it is impossible, literally impossible, to say with absolute certainty what any English utterance means exactly if we do not have some context. The same is almost certainly true of all other languages.