There are no mathematical reasons. It is a question of convenience. If you use the first definition, the inverse transform will have $1/2\pi$. If you use the third definition, the inverse transform will have $1/\sqrt{2\pi}$ (and will be very similar to the direct transform). If you use the second definition, none will have any $2\pi$ in front. So the convenience of the second and third definition is a symmetry between the direct and inverse transform.

Finally your list misses one more possible definition used in some books: with $1/2\pi$ in front of direct transform. Then the inverse one has no multiple in front.

To conclude: these $2\pi$'s are unavoidable, where to place them is a matter or taste and convenience.

There are no mathematical reasons. It is a question of convenience. If you use the first definition, the inverse transform will have $1/2\pi$. If you use the third definition, the inverse transform will have $1/\sqrt{2\pi}$ (and will be very similar to the direct transform). If you use the second definition, none will have any $2\pi$ in front. So the convenience of the second and third definition is a symmetry between the direct and inverse transform.

Finally your list misses one more possible definition used in some books: with $1/2\pi$ in front of direct transform. Then the inverse one has no multiple in front. (For example, the textbook by Folland, Fourier Analysis and its applications).

To conclude: these $2\pi$'s are unavoidable, where to place them is a matter or taste and convenience. Same applies to Fourier series, of course. I've seen options 1, 2, 3 in textbooks.