The version 2 is also popular among electrical engineers as the variable $\xi$ is then the actual **frequency**. For an electrical engineering view on the Fourier transform, I can recommend the lecture notes [The Fourier Transform and its Applications][1] by Brad Osgood.

Also, this notion of frequency explains that electrical engineers sometimes use variable names that seem a bit odd for mathematicians: A (complex valued) signal with angular frequency $\omega$ is $\exp(i\omega t)$ and written in linear frequency $f = \omega/(2\pi)$ it is $\exp(i2\pi ft)$. Hence, the Fourier transform of a signal $x(t)$ ($t$ in seconds) may look like $\hat x(f) = \int_{-\infty}^\infty \exp(i2\pi ft)x(t) dt$ ($f$ in $\mathrm{Hz}$).

One downside of this convention is that the handy rule of differentiation gets a bit more complicated (namely $\hat{x'}(f) = 2\pi i f\, \hat x(f)$ instead of $\hat{x'}(f) = if\,\hat x(f)$).


  [1]: https://see.stanford.edu/materials/lsoftaee261/book-fall-07.pdf