I agree with your opinion about the naming being confusing. Specifically regarding your mathematician friend what would you lose by taking a fast Fourier transform over a normal Fourier transform? Well the two aren't interchangeable. You would lose continuous time / frequency!

Some personal preference:

I find it hard to read the grey text on a white background that you have, and it's probably just a fundamental limit of reader mode in firefox, but it doesn't render mathml right. To read it I zoomed in, but then there were CSS issues where the content overlapped the sidebar.

While |x| is common to reference the length of a set I've not really seen that to reference the number of elements in a vector in the fields where I've used discrete Fourier transforms. I've always just defined N as the length of my vector. I honestly read it at first as the norm of x, and the norm of F{x} and thought you might be about to talk about Parseval's theorem.

Enjoyable enough and accurate article though. Thanks!