AS for your example of LAPACK, "The existence of evening, my dear Boswell, does not mean that day is the same as night." Numerical linear algebra is on a boundary between mathematics and computer science. There is a large boundary between math and computer science. But then again the same can be said of, say, math and physics. But by the time you're worried about how to use caching effectively, and make that work in a distributed computation, you're pretty far on the computer science side.
Also your idea of grouping them because of research methods is problematic at best. For example machine learning research has a highly experimental flavor that does not fit in mathematics at all. Quantum computing contains a lot that is closer to a field of applied physics than of mathematics. And so on.
I honestly don't see how that follows. Does the fact that writing LAPACK required software engineering mean that numerical linear algebra is not a part of mathematics?
My basis for grouping them together is because of the subjects under study and research methods used. You can always claim that two different fields are different, but I think making arguments for some relationship is more useful.