The right question is "what's the difference between a bug-fix and a breaking change?"
A bug fix is a change. A breaking change is a change that breaks the API. Doesn't matter if the previous status was the intended one or not.
SemVer not working is almost entirely people simply not following it correctly. The projects that follow it as best as they can have very few mistakes were "woops that was a major change" occurs while projects that use it as a guideline may as well not be using it at all.
Often, there are ways to work around bugs in an API. And, just as often, those workarounds will break as soon as the bug is fixed. The API developer is in a particularly poor place to judge whether a bug fix is breaking or not. Sometimes, they can talk to users to see if a change will break their code, but other times they can't. Thus, it comes down to how well a developer can predict whether a fix will break projects in the wild.
No, it isn't. The most relevant question is "can somebody be using the current API". It doesn't matter if your current API matches the documentation, what matters is whether your current API is out there for others to build on.
Don't try to use a crystal ball or other form of divination to predict what your downstream users have been doing with the code; you will always lose. Instead, suck it up, acknowledge the mistake, and signal the breaking change by bumping the major version.
Maybe next time your developers will spend more time validating their public contract, so they won't have to endure the embarrassment of a major version bump.
Semver both acknowledges and is predicated upon the fact that every bug fix has a range of possible impacts. At one end, some bug fixes have almost no chance of breaking an application. At the other end, some bug fixes will almost certainly break applications.
This is not divination. It is understanding your application and its users.
Finally, in regards to your last paragraph, I genuinely hope that you don't speak to developers like that. Mistakes will always happen. Have a little bit of respect and treat people with kindness.
Yes. That's exactly how SemVer works. It signals to people that there was a breaking API change - it doesn't matter if you're on version 10495.0.3 - if you've somehow broken your API 10,494 times you're following SemVer. Although if you're that bad at maintaining a stable API you probably don't have any users you know... using it.
SemVer is predicated upon the fact that other people are using your API. It signals to them whether or not it should be safe to update without breaking their code that relies on your API. If you break your API - you broke their code. That makes you an asshole in their eyes. The entire point of SemVer is to signal that you broke your API so that people relying on it can examine their code for breakage, decide whether or not to update, or start working on modifying their code to match the new API.
If your API is very unstable then people aren't going to rely on it. If it's both unstable and you don't signal breaking changes I'd be amazed if anyone put up with working with it.
If I'm using version 3.0.2 I should be fine using any 3.y.z version. If you release a 4.0.0 then I know I need to investigate and determine whether or not it is safe for me to update to that version. If your API is still unstable and changing on a daily basis - you shouldn't be at 1.0.0 yet. 1.0.0 is meant to signal the "stable API stage" of development.
Doesn't matter if it breaks it in a way that you don't expect users to have been using it - you broke it. Doesn't matter if you were just fixing a typo that changed behavior - you broke it. Doesn't matter if you were fixing a logical error - you broke it. Doesn't matter if you fixed the PRNG causing some results to no longer be possible - you broke it. Doesn't matter if you deleted a deprecated endpoint - you broke it. '
SemVer isn't asking why you broke it - it's asking did you break it.