FFT is simply an algorithm to efficiently compute the DFT. The fact that the article makes no mention of DFT vs fourier series vs DTFT is going to end up creating more confusion that it solves. For some reason introductory tutorials always start with the DFT (usually mistakenly using FFT and DFT interchangeably), even though to me the continuous fourier transform is far easier to conceptually understand. Going from continuous fourier transform to the DTFT, is just applying the FT to a dirac-combed (sampled) function. Then from DTFT to DFT you introduce periodic boundary condition. Fourier series is just applying FT to a function that happens to already be periodic, resulting in a finite set of discrete frequencies.
There is a connection between fourier series and DFT in that if the fourier series is computed for the periodic resummation of a signal, and then the DFT is computed for the original signal (which implicitly involves applying a periodic boundary condition), the DFT is just the periodic resummation of the fourier series.
I spent ages meditating on this image https://en.wikipedia.org/wiki/Discrete_Fourier_transform?#/m... before everything finally clicked, it's a shame that introductions never once mention DTFT