I'd also argue that statistical mechanics and biophysics applications are missing in the list above-- both use fairly specific mathematical tools in some unusual ways.
Actually, when you get right down to it, every STEM field has its own version of applied mathematics, and in the case of biochemistry, chemistry, and physics, multiple subdomains. There is certainly overlap with pure mathematics, and applications from other fields, but each domain has its own thing.
For example, the geometry and modeling that goes into receptor-ligand models and their kinetics/mechanisms. Or protein folding. Or screening libraries for structure-activity relationships.
Most awesome collision of fields ever for me as a polymath, by the way. My spouse would say the same thing about thin-film materials science instead, though, so there are a LOT of those little niches in STEM.
I realize that this gets away from "pure" mathematics, but honestly, most of the other things aren't pure mathematics, either, when you get right down to it.
Being able to construct a model that predicts market fluctuations on the basis of a set of inputs isn't that different from being able to construct a model that predicts the diffusion of a radioactive contaminant that is partitioning between soil particles and moving groundwater. KWIM?
