You are thinking about the weeds of sorting now, but that is rarely relevant to the problem at hand.
You can (and most people do) look at it from the other angle. Sort(seq) is a function which returns a permutation of seq such that every element is less-than-or-equal-to the next element. Usually, we use this function because we need this property to hold in our following code. A lot of the time, this function is called for very short sequences, where even the most naive implementation would be fats enough for even the highest scale program. So, in the vast majority of code we only care about the semantics of std::sort; and all of the above have the same semantics.
In fact, the fact that we do often get a stable sort as well shows that semantics are the most important characteristic, as stable and unstable sorts do have slightly different semantics.
You can (and most people do) look at it from the other angle. Sort(seq) is a function which returns a permutation of seq such that every element is less-than-or-equal-to the next element. Usually, we use this function because we need this property to hold in our following code. A lot of the time, this function is called for very short sequences, where even the most naive implementation would be fats enough for even the highest scale program. So, in the vast majority of code we only care about the semantics of std::sort; and all of the above have the same semantics.
In fact, the fact that we do often get a stable sort as well shows that semantics are the most important characteristic, as stable and unstable sorts do have slightly different semantics.