Both are "correct", provided that in your first paragraph one of the =
signs is actually a +
sign.
The basic problem is: there isn't just one single definition of the Fourier transform! The Fourier transform of a function $f(t)$ generally looks like this
$ \mathcal{F}f(\omega) = A \int_{-\infty}^{\infty} f(t) \exp ( - C i \omega t) \mathrm{d}t $
where $C$ and $A$ are fixed non-zero constants. Common choices of $C$ includes $1$ or $2\pi$. While common choices of $A$ include $1$, $1/2\pi$, or $1/\sqrt{2\pi}$. Wikipedia has some more discussion about this. The different choices of $C$ and $A$ have different desireable properties. If $C$ is set to be $1$, then under the Fourier transform the derivative $\frac{d}{dt}$ becomes $-i$. However, the frequency $\omega$ should be then interpreted as an "angular frequency" instead of a frequency in terms of signal processing. Conversely, setting $C$ to $2\pi$ you get that $\omega$ is an actual frequency, but the relation with the derivatives pick up extra factors of $2\pi$.
Similarly, setting $A$ to different values makes the Fourier inversion formula look different. Some of which are intuitively more natural when you consider the Fourier transform as breaking down a signal into sines and cosines, and some of which are more "aesthetically pleasing" (the choice of $A = 1/\sqrt{2\pi}$ makes the Fourier and Fourier-inverse transforms look more similar to each other).
In short, whenever you consult a reference, you should first double check the convention the author uses for the Fourier transform. Between different conventions there will be different factors of $2\pi$ floating around. And as long as you keep track of them and make sure you only write using one fixed convention, you shouldn't run into problems.