optics - Diffraction and $k$-space

Regarding diffraction I am a little bit lost reading about reciprocal space and the space of $k$'s. As I understand it the Fourier relationship between a wavepacket $\Psi(\vec r,t)$ and the complex weighting factors of each constituent plane wave $A(\vec k)$ is given by: $$\begin{equation} \Psi(\vec r,t)=\frac{1}{\sqrt{2\pi}}\int ^{\infty}_{-\infty}A(\vec k)e^{i(\vec k\vec r-\omega t)}d\vec k \end{equation}$$ demonstrating a sort of linear superposition of reflected plane waves from a diffraction grating (or crystal lattice).Further by Parseval's theorem the intensity of this reflected packet is given by: $$\begin{equation} \int^{\infty}_{-\infty}\big|\Psi(\vec r,t)\big|^2d\vec r=\int^{\infty}_{-\infty}\big|A(\vec k)\big|^2d\vec k\end{equation}$$

However I am not really sure how this relates to the other sort of understanding of $k$ space. That is to say the space that can give us meaningful information about crystal lattices and unit cells. Are they the same spaces?

Would this mean therefore that the intensity/position of the diffraction spots can be related to the structure of the solid's lattice. If so how can we understand distributions in terms of the Fourier relationship above?

I understand there have been several questions so far regarding the reciprocal k-space however so far I have not found anything that helps me particularly grasp this aspect of diffraction.

As you can see I am quiet confused in this matter and would greatly appreciate some help!