The sense of that

[Progresser]

Hello teachers. I am new here, Japanese male and sincere English learner.
I would like to acquire the sense of English language, not the translated words.
Please let me ask native speakers of English the sense (meaning, feeling, core) of the word that.

1. That is my book. (pron.: pointing out and directing your attention to a real book )
2. I was living with my parents at that time (det. : the time already mentioned, so pointing out and directing the listener’s attention to their memory)
3. I like the book that you gave me yesterday. (relative pron. : pointing out and directing the listener’s attention to the book)
4. I believe that you'll get on in the world. (conj. by saying that, directing the listener’s attention to after that and make it possible to take the that clause as one group.
5. It’s possible that he has not received the letter.(when the listener hears “it’s possible”, they think what is possible. Next they hear that , that directs the listener’s attention to after that and points out the content of it.
6. It’s he that broke the vase yesterday. ( emphasizing structure: when the listener hears “it’s he”, they think what is he, and hears that the same as above)
7. I like the idea that we will throw a party for her. (conj.: they think what kind of idea? That can directs the listener’s attention to after that and point out the content of the idea)

I wrote seven sentences to show my question clear. My question is that that is used or means to direct the listener’s attention to a real thing, memory, the previous word or the sentence after that.
Each grammatical function is different, but its basic sense is the same. Is it correct?
The reason why I asked of that is that we have to use its suitable word for translation in each case. Many learners are confused for that. Incidentally, I understand every single grammatical function, meaning and translation but the sense. Thank you vey much for reading my question.

[Raymott]

Hello, welcome to the forum.
For me, the word 'that' doesn't have a sense or a feeling. But if I had to nominate the essence of 'thatness' it would be "a reference to something else". But that's purely subjective.