You are correct about private interfaces. When I wrote that I was imagining something more significant like a code refactor, but yes, obviously, changing something like an internal function definition would not require a new major version if it doesn't change a public interface. Similarly, implementing a bugfix or new feature wouldn't necessarily mean that an existing public interface was broken, or that the major version should be incremented. I didn't intend to imply that.
I am using public interfaces in my examples because the original point was how SemVer can communicate at a glance to the end user the kinds of changes that were made (compatibility-breaking, bugfix, etc.) and I had the offhand idea to also communicate when the update was released by including the date in the patch number. I am not confused about what semantic versioning is or whether it can only apply to public interfaces or libraries. If I knew it was going to start an argument, I wouldn't have mentioned backwards compatibility; it was an offhand comment tangential to the idea I was explaining. I could have just as easily said:
"I prefer the SemVer
Major.Minor.Patchapproach so I can tell at a glance if the update is a new feature release or is just bug fixes".
I don't think I skipped the question about Firefox interfaces. An interface I was looking at for backwards compatibility was in the example I provided with the UserChrome interface and I provided a specific example of a third party tool using that interface, the FireFox-UI-Fix project. Admittedly, this isn't a strong example because the UserChrome customization doesn't expose any functions to be called and doesn't define any kind of protocol in a traditional sense. But that doesn't make it any less of an interface in my opinion.
The UserChromeCSS customization feature is provided to the user by Mozilla for the purpose of modifying the browser's chrome i.e. graphical user interface (note I'm not confusing a gui with a programming interface, they just happen to be the same thing in this example). In order to make these customizations, the user must be aware of how the browser's gui is layed out, i.e. the user must know the structure of the HTML that makes up the browser's chrome. If the user writes a gui customization which depends on that structure for one version of the web browser but then the browser changes that HTML structure in the next update, that constitutes a breaking change. In this example the interface is defined by the chrome's HTML itself. The CSS written in the UserChrome.css references/selects that HTML and is thus dependent on the stability of that HTML in order to produce the same effects across different versions of the web browser. Third-party tools that distribute custom UserChrome.css files should therefore expect that their customizations be compatible across minor and patch versions of the same major version release. It's not necessarily that the major version must increment every time this gui is changed, but when the interface for customizing this gui has introduced a breaking change (which in this case is usually synonymous). I think this is what you mean when you say "userflow". If so, then no, I don't think "userflow" is an interface. The userflow/gui happens to be an interface in this example because of the UserChrome feature that exposes the gui to modification through its own HTML/CSS interface, the stability of which is depended upon by both users and third-party developers such as the Firefox-UI-Fix project I mentioned.
As for other Firefox interfaces which would call for a major version increment upon being changed, there is the WebExtensions API for browser extensions, and the cli arguments that you mentioned. I don't think providing an exhaustive list supports or invalidates any point or opinion I've stated. The major version number is incremented if any public interface changes, it doesn't have to be representative of a single interface exclusively. An application can provide multiple public interfaces, where a library tends to be more singularly focused (maybe this is the source of our disagreement/misunderstanding?). An incremented major version just means that there is some breaking change(s) in some interface(s). Conversely, an incremented major version number doesn't imply that every provided interface contains a breaking change.
If it's your opinion that SemVer is better suited to a narrow API or library where a new major version exclusively indicates a breaking change in its singular public interface. Ok. That doesn't indicate a lack of understanding SemVer on my part, and that's not a requirement of SemVer. There exist applications using SemVer that expose multiple interfaces.
The claim wasn't that a code refactor is always a change in public interface, but that it could constitute a new major version. I listed two examples of when a major version should be incremented, the first being a change in a public interface, the second (erroneously) was a change in a private interface which I then clarified could only apply in the case of a more substantial code refactor, because as you pointed out (and I reiterated and agreed with), private interface changes don't necessitate breaking changes. It isn't an exclusive requirement that a public interface has breaking changes in order for the major version to be incremented, only that there be a new major version when that interface ~~breaks~~ introduces breaking changes.
I had to explain userchrome thoroughly in order to demonstrate that it is a public interface and differentiate it from the gui. I assumed it wasn't intuitive because you missed it when I provided it as an example initially and was accused of avoiding that point.
The first sentence of each paragraph addresses which point it argues other than the userchrome demonstration which follows from the prior paragraph and only addresses your userflow vs interface question in its conclusion.