Well, I suppose we could get into a big philosophical discussion about this, but to explain myself, in my usage, a “fact” is an indisputable logical or scientific proposition about reality. “Truth” is a broader category for the “really real” and includes things we come to know through experience, imagination, revelation and other sources of knowledge besides reason and observation. Many “truths” are disputable or nuanced and depend on your view of reality, whereas facts can be proven true or false.
So I do not believe the words are synonymous or that there is mutual entailment, they are a hyponym/hypernym pair. A fact is a kind of truth like a dog is a kind of animal. If something is a fact, it entails it is also a truth, but to claim you can therefore negate q and entail p is a logical fallacy. (If a creature is an animal, it doesn’t entail it is a dog.) The entailment only holds if you negate the consequence and make it the premise. (If something is not a truth, it is therefore not a fact.)
When you say “history” it is obvious you mean objective recording of objective facts. That is not what “history” means in many cultures. Genealogies play a role in many cultures of marking a text as authoritative and legitimate. Yes, the genealogies in Genesis tie the narratives to the history of Israel. That is partially why I believe they talk about people who really lived. But it is an unwarranted leap to say that because the narratives include characters who lived in history, therefore we should read and interpret them as we read modern history, with an expectation that the composers were following our modern conventions for describing reality or that their reasons for composing the histories were simply to accurately inform posterity of what happened. Even in our own day, histories are told from a perspective and serve many cultural purposes other than (or in addition to) recording facts and informing. And the ANE motivations and conventions were obviously different than our own. The numerology of the recorded life spans is one example.
My sympathies, you probably have a lot of baggage to unpack. I think it is relevant to how you understand Christians who answer your questions. Over the course of many discussions with people who have grown up in this subculture, I have noticed that there seems to be an awful lot of “either it has to be this, and if it is not, it must be this” kind of thinking. A huge percentage of these assertions are false choices and don’t represent the way other Christians (who in my opinion have a more sane and healthy way of approaching God, truth, and the Bible) think about an issue. So it is something for you to be aware of when you are trying to understand what people here are saying. Isaiah 53 is a case in point. The idea that the main reason it is in the Bible is to act as an apologetic tool to prove the legitimacy of Christ seems to me to be a weirdly exclusive way to approach the text. “Either the OT is fact or the OT is fiction” is another false choice that is pretty incomprehensible to me.