I don't understand with the below equations the affirmation that $FT(\Delta(\vec{k})$ is the Fourier transform of $\Delta(\vec{r})$ :
$$\left\{\begin{array}{l}{\Delta(\vec{r})=\frac{V}{(2 \pi)^{3}} \int \tilde{\Delta}(\vec{k}) e^{-i \vec{k} \cdot \vec{r}} d^{3} \vec{k}} \\ {\tilde{\Delta}(\vec{k})=\frac{1}{V} \int \Delta(\vec{r}) e^{i \vec{k} \cdot \vec{r}} d^{3} \vec{r}}\end{array}\right.$$
with the contrast of density :
$$\Delta(\vec{x}) \equiv \frac{\rho(\vec{x})}{\rho_{0}}-1$$
But classicaly, (Forward) Fourier transform is defined by:
$$FT(\Delta(\vec{r}))=\Delta(\vec{k})=\dfrac{V}{(2\pi)^3}\,\int\,\Delta(\vec{r})\,e^{-i\vec{k}\cdot\vec{r}}\text{d}^3\vec{r}.$$
So there should be a plus sign at the beginning of post for the first formula in top image (Inverse Fourier Transform) and I expect a minus sign for the second formula. ( I didn't mention also the difference of factors between the 2 formula but this is not my main issue).
You're right that in my experience, I have found most authors use the minus sign for the forward transform. However, the important thing is that the forward and inverse Fourier transforms have opposite signs in the exponent; which one is positive and which is negative is ultimately a matter of convention, in the sense that
$$\Delta^+(\vec k) = \int d^3 \vec r \ \ \Delta(\vec r) e^{i\vec k \cdot \vec r}$$ and $$\Delta^-(\vec k) = \int d^3 \vec r \ \ \Delta(\vec r) e^{-i\vec k \cdot \vec r}$$
contain precisely the same information. For instance, if $\Delta(\vec r)$ is a real-valued function, then the two transformed functions above are just complex conjugates of one another, so as long as you are consistent in your definitions then you could work with either one.
This is similar to the fact that if you transform forward and then backward, you need to include an overall factor of $\frac{1}{(2\pi)^3}$. Whether you attach that to the forward transform or the inverse transform, or assign each of them a factor of $\frac{1}{(2\pi)^{3/2}}$, is a matter of convention as well.