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?
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.