For example, with respect to compressibility, the author isn't saying that water isn't compressible in the physical sense, but rather that it's not compressible in the mode of its behavior in their prior example (which they refer to as a "Naïve Implementation"). Which they're correct about -- their naive implementation would be more like an ideal-gas at near-vacuum pressures, which would be pretty unrealistic for water under normal conditions. And they also explain that non-compressibility (in the physical sense) is merely an approximation.
Then as for pressure-differences, the author has stuff like a variable called "difference_in_p" and writes about the pressure-field and such. So, no worries, they're not actually confused about incompressible fluids being able to have pressure-differences.
You are right that they typo'd "Navier-Stokes equations" in two different ways: they always seem to miss the last "s" in "Stokes", and sometimes they non-idiomatically put Stokes's name before Navier's. Presumably the author should fix the spelling of Stokes's name -- the latter might be a stylistic quirk (sometimes folks may intentionally re-order a list of names, arguing that it'd be more fair that way).
All that said, I can appreciate how it might be odd to see common terms used in a conceptual-development in a manner inconsistent with their meanings in the standard literature. It might be helpful for the author to include something like a disclaimer that they're taking a constructivist approach, and that the terminology used during construction may vary from the final definitions one'd arrive at once the standard theory is constructed.