The author probably missed that the first F in FFT refers to a very specific (but efficient) algorithm, and that the effects he achieved are due to the properties of the (2D) Fourier transform, which can be computed using other algorithms as well.
Completely correct. And yet, FFT is much less ambiguous than FT ("Foot"? "Financial Times"? "Face time"?), so imo what is lost in preciseness is gained in clarity.
Discrete Fourier Transform is probably the most accurate way to describe the mathematical transformation which is being applied. Albeit it is very popular to say FFT when what is meant is Fourier transform or DFT, it is a bit like talking about sorting but calling it Quicksort despite it being a specific algorithm.