These two pairs of words are indeed quite similar, and can sometimes be used interchangeably, but the connotations differ enough that usually these terms are distinguishable. Generally speaking, to accept something is a positive action, implying that you are at terms with whatever you are accepting and are, more or less, embracing it -- on the other hand, to admit something directly implies reluctance. This reluctance, or lack thereof in acceptance, has implications.
Similarly, there is a difference between you failing and you being defeated. To fail implies that the fault most likely resides in yourself, and it carries a more negative connotation. Meanwhile, defeat is a more generic term -- the difference here is more subtle, so allow this example:
a) The majority party's candidate was defeated in the primaries.
b) The absence of the majority party in the election is due to the failure of their candidate, who was defeated in the primaries.
Heavy connotative differences can surface, if one knows how to manipulate these terms. So, with these two pairs of terms each with slightly different connotations, the combinations exacerbate these effects. Now, onto your examples:
I thought I could change her attitude, but she was too dogmatic to be convinced about her beliefs. I had to admit defeat.
I wouldn't call this a failure, as the narrator was never tasked with changing her mind, nor was this a serious matter -- the lack of importance makes failure a term too strong, I would argue. As for admit/accept, because the author realizes that they were proven wrong, this sounds more like an admission of defeat rather than an acceptance. This one is a little harder to explain, forgive me.
They were arm-wrestling; George was much stronger than Stephan; George was laughing and didn't try too much, while Stephan had turned red and was struggling a lot to not lose. I think he was too proud to admit defeat.
This case is a little special, I would say, as in sports the term defeat is used far more often than failure, at least when talking about the performance of competitors. Losing a game, even an unofficial arm-wrestling exhibition match, is not generally denoted as a failure of the players. On top of this, Stephan was directly stated as being proud and unable to accept the inevitable outcome of this event -- therefore, this reluctance suggests the usage of admit. While I chose the same combination for both of these examples, it is certainly not the only viable one.