Union of differences between three sets, where nothing intersects, how is it called?

I don't know of a name for this, but one way to express it would be

$$\bigcup\limits_i A_i \setminus \bigcup\limits_{i\ne j} \left(A_i \cap A_j\right)$$

You don't need to specify more complicated intersections, because they're already included in what's being subtracted here. This captures the idea that it's intersections that you don't want.

This also provides for a succinct description of it in words: "the union minus the pairwise intersections".

This is similar to, but distinct from the symmetric difference.

If your collection is $\mathcal{C}=\{A,B,C,\ldots\}$, then you could write this set with cumbersome notation along the lines of $\left\{a\in\bigcup \mathcal{C}\left|\,\left|\left\{A\in\mathcal{C}\mid a\in A\right\}\right|=1\right.\right\}$. It would probably be clearer to use words, like "$\nabla \mathcal{C}$ will denote the set of all elements that are in exactly one set in the collection $\mathcal{C}$.