(these kinds of) Crashes can be manipulated, though. If you can't restore a stable state, you have created instability. Failing gracefully in this context means failing without creating instability or insecurity.
An abort()/__builtin_trap/etc shouldn't generally be manipulable.
The failure mode is the one where you terminate in the middle of an operation that was expected to be atomic or ordered and then the program restarts and does step two without finishing step one.
You have to protect against that anyway, otherwise you're vulnerable to power cuts.
And operating system crashes, and being killed by the OOM handler (which is probably likely if you're out of heap), and all of the other things that can kill a process without warning.
Well, considering malloc gives a NULL pointer on failure, a crash at some point is pretty clearly inevitable.
Yes, but it becomes harder to tell where in the code that happened. If you run abort() or the like, you've got more information as to the circumstances of the crash.
Sadly it doesn't on linux due to overcommit. You can actually get the failure much later when you try to use the memory. The exception is when you try to allocate absolutely huge buffers. So malloc returning a valid pointer doesn't indicate success (though it returning NULL does indicate failure).