Just sharing an anecdote: recently, I had to create Linux images for x86 on ARM machine using QEMU. During this process, I discovered that, for example, creation of initrd fails because of memory page size (some code makes assumption about page size and calculates the memory location to access instead of using system interface to discover that location). There's a similar problem when using "locate" utility. Probably a bunch more programs that have been successfully used millions, well, probably trillions times. This manifests itself in QEMU segfaulting when trying to perform these operations.
But, to answer the question: I think, one way to define memory safety is to ensure that the language doesn't have the ability to do I/O to a memory address not obtained through system interface. Not sure if this is too much to ask. Feels like for application development purposes this should be OK, and for system development this obviously will not work (someone has to create the system interface that supplies valid memory addresses).
- out-of-bounds on array read/write
- stack corruption such as overwriting the return address
It doesn't directly say "you can't use C", but achieving this level of soundness in C is quite hard (see sel4 and its Coq proof).
But there's another aspect that I think you missed: use after free.
As you say, achieving this level of soundness with C is hard. Proving it is much harder. (Except, how do you know you've achieved it if you don't prove it?)
Also these properties are a very small subset of general correctness. Who cares if you write a "safe" program if it computes the wrong answer?
Err .. that is actually the point of the proof. Can you give an example of something with a Coq-type safety proof that has a memory safety bug in it?