Note that this also makes for example the uppermost part of the JVM an OS by my definition. I’d argue it’s not a completely incorrect characterization.
For too narrow: you might have a security focused OS that includes everything like a TCP/IP stack etc, but perhaps doesn't allow arbitrary binaries by construction. (Eg it might only allow binaries that come with a proof of innocence, or some other restriction that's harsh enough to lose the 'arbitrary' rating.)
For too broad: https://en.wikipedia.org/wiki/Weird_machine would make almost any software that was written sloppily enough into an OS.
Imagine some embedded application, eg in car, that has a network stack, graphics drivers etc, but can only run built-in functionality.
In your example, a real or virtual machine’s load/setregs interface is an operating system. One that can only run a single program at a time in the case of the physical processor’s bootstrap, but you’d be surprised how similar that is to early DOS.
A more hand-waving version is “the lowest level interface over the machine that can run...”. So a processor and a ROM grid can run arbitrary programs, but the lowest common denominator entry point is the ROM flashing spec! A very slow scheduler indeed.
No, every operating system that I know of makes some assumptions about the language that programs are written in. On my amd64/Ubuntu system, the language is largely specified by the processor (and, yes, has a binary format) and the programs are allowed to assume the presence of various devices, formats of various system calls, etc. Other languages are supported through a complicated system of compilers, interpreters and assemblers.
Or, coming at a different tack: does the existence of a C->Smalltalk transpiler turn the Smalltalk-based pseudo-OS into a real OS? What if the pseudo-OS doesn't ship the transpiler, and you need to run that yourself in user-space?
For example, you can have a Smalltalk-based OS, written in Smalltalk. The OS components are simply Smalltalk classes and methods. Your programs/utilities/applications would also be Smalltalk classes and methods. There needn't be any clear boundary between the classes/methods that belong to the OS and those that belong to programs/applications/utilities.
If you look at library operating systems, the operating system is just a library which you link into your application. Such an operating system can only run one program, and you have to rebuild the operating system whenever you want to change the single program it runs. Still, I think library operating systems count as operating systems. This Tetris implementation isn't using a library operating system, but you could refactor the code into two parts – the Tetris game, and generic OS services, and the later would constitute a library operating system. If you look at its source files, only about seven of them (main.c, music.[ch], sound.[ch], speaker.[ch]) actually implement the game. The other 23 source files are generic OS services. So I'd say that, even if this isn't an operating system as a whole, it contains an OS within itself.