Could you try to provide an illustrative example?
For an English person, maybe the best example is what is called Mankading, mainly because it involves cricket. An incident of Mankading occurred in a test match (= international game) between India and England in (I think) 2019. It caused a huge fuss and made national headlines, even though it was not the first time it has happened (the original incident involving Mankad was about 70 years ago).
It will probably be hard for almost all non-English or non-Commonwealth people to understand, but I think it's the kind of thing worth focusing on because it does illustrate what I regard as distinctly British aspects of the concept.
I assume you will have some idea of baseball. Imagine a pitcher throwing a ball at a batter. If the batter hits the ball safely, he can run round the bases. To switch this to cricket, imagine another batter stands beside the pitcher (the bowler in cricket), and that at each better's end is a set of stumps - a wicket - three short poles stuck close together in the ground. If the batter hits the ball, he can then run, NOT round the bases, but to where the bowler bowled from. The other batter has to run to where the first batter started from, and if they both safely reach the respective opposite positions, a run is scored (the other batter would face the bowler for the next play. But if a player on the fielding side (including the bowler) "breaks" the wicket with the ball before either batter has safely reached home, the runner at that end is out.
Now what happens in practice is that, as the bowler is delivering the ball to the first batter, the second batter typically starts walking towards the first batter's end, so that if the ball is safely struck, he has a head start in reaching the other end safely. Essentially, he is cheating. (It is a bit like stealing a base in baseball.)
What Mankad famously did as the bowler in this situation was to stop and not deliver the ball. Instead, he broke (hit) the wicket with the ball at his end, and so batter at his end, who had gone walkabout, was stranded between the wickets. He was out. There is absolutely no doubt about this. Everyone agrees 100%.
But it causes a huge fuss in England because it is considered not fair play by the bowler. In India and Pakistan, where Mankading is common, it causes a mere ripple of protest. Different culture. The majority English view is that the correct procedure is for the bowler to stop and simply warn the walkabout batter, effectively saying, "I don't want to impugn your honour, but I'm sure you wouldn't want to give the impression you might be cheating. So, I can dismiss you but I won't this time. Next time, maybe..." That's considered fair-play. No rule book or debate involved. The sympathy is with the runner. The bowler is lambasted.
Consider a similar situation with stealing a base in baseball. The pitcher does not deliver his pitch to the batter but throws the ball to the first-base fielder because the runner there is trying to steal a march to get to second base. If the runner gets back to base before he is tagged, nothing happens and the fielder just throws the ball back to the pitcher. Except that sometimes he doesn't. He makes afake throw back to the pitcher and keeps the ball hidden in his glove. The runner relaxes and steps off the base, but then the fielder surprises him by tagging him out with the ball. In America there is NO sympathy for the runner. He is just seen as a doofus, or a twit. The first-base fielder is applauded. Different culture. And it doesn't make the news.
For example: I have witnessed several times how players ignore that their opponent has run out of time in byouyomi.
I'm not quite sure what you mean by this, but if you mean that the player who has time on his clock
knowingly lets his opponent play on (and so does not claim a win on time), I wouldn't count that as an example of (British) fair play. I have seen it many times. I would class it simply as an example of good manners, noblesse oblige or good sportsmanship, depending on the context.
Since you mention byoyomi, there is (or used to be) a good example there of Japanese fair play (which I regard as very close to the British sense). When a senior player is in byoyomi, the young timekeeper will not count out the seconds metronomically 10, 9. 8. 7.... He will count VERY slowly: 10..........., 9........, 8...... etc. And I have seen no case where the opponent objected. The very, very few examples I can recall where a player did lose on time involved other factors. For example, Suzuki lost on time to Segoe but that was almost certainly because Suzuki had irritated ALL other players by insisting on 16 hours each instead of the usual 11 hours each.
The whole point of fair play has to be that there is no rule book that tells you what fair play actually is. It is doing whatever is considered to be in the "spirit of the game" in that particular culture and at that particular time. Just as there is no law that says gentlemen have to lift the seat. It is just the done thing.