Different people have different ideas of what constitutes “fairness” in different situations. That much is obvious.
But that has no bearing on the general meaning of the word “fair”, just on its interpretation in specific contexts.
In a particular scenario, we can say a solution X is proposed which Alice believes is “fair” but which Bob does not believe is “fair”. Given your proposed meaning for the word “fair”, X must be fair because Alice believes it to be fair. But by the same token, X must be unfair because Bob believes it to be unfair and there’s no obvious a priori reason to prefer Alice’s subjective sense of fairness to Bob’s.
Thus, your definition of “fair” entails a logical contradiction in many possible scenarios, implying that it is insufficient to capture the actual meaning of the word “fair”. The problem seems to be when you say: “they are usually determining what is fair to themselves and the persons they care for”. Fairness precludes individuals determining what is “fair” – the meaning of “fair” must be determined either by an external set of rules or criteria for fairness or some other disinterested arbiter or mediator.
So I respectfully disagree with your definition of the word “fair” and maintain that the definition given by googling the word is better, and will continue to use it in that way until such a time as someone provides a better one.