As far as I understand, the coend of a diagram $X:I^\text{op}\times I\to D$ is an object $x\in D_0$ together with a natural isomorphism $\alpha:[[I,D]](X,\Delta^e_*)\cong D(x,*)$ in $[D,\text{Set}]$ where I denote with $[[I,D]]$ the category of diagrams $I^\text{op}\times I\to D$ and extranatural transformations between them as well as with $\Delta^e_*:D\to[[I,D]]$ the functor taking an object $y$ to the constant diagram $(i_0,i_1)\mapsto y$.
I feel now a bit stupid because I cannot tell what the analogous concept for natural transformations is called. I'm sure I have already seen it but, given $X:I\to D$, what exactly is an object $x\in D_0$ together with a natural transformation $\alpha:[I,D](X,\Delta_*)\cong D(x,*)$?
Also, what about the notion $\int^i X(i,i)$? The coend does not only depend on the $X(i,i)$ values, right?
The case for ordinary natural transformations is just that of a colimit. The natural isomorphism $[I,D](X,\Delta_*)\simeq D(x,*)$ says that there is a natural correspondence between morphisms $x\to y$ for any $y$, and cocones under $X$ with vertex $y$ (cocones are just natural transformations to constant functors); and this is just what it means to be a colimit.
And what's important in the notation $\int^iX(i,i)$ is that it binds those variable positions--it is not saying that it is only constructed from the "diagonal" values. It can be important to be clear on what variables you're binding because it might happen that you have a diagram $F:C\times C^{op}\times C\to D$ and can form the coend $\int^cF(c,c,c')$ or $\int^cF(c',c,c)$ for a given parameter $c'$, and there's no reason these will be the same things; the notation tells you which two arguments you're taking the coend with respect to.