I've noticed that "uncorrelated" is synonymously used for "pairwise uncorrelated" even when more than two random variables are considered.
Maybe I'm missing something, but this seems strange to me. We know that (mutual) independence and pairwise independence are not the equivalent if more than two random variables are considered. Pairwise independence implies pairwise uncorrelatedness, but (mutual) independence of $X_1,\ldots,X_n$ even implies $$\operatorname E\left[\prod_{i=1}^nX_i\right]=\prod_{i=1}^n\operatorname E\left[X_i\right]\tag1$$ which is strictly stronger than pairwise uncorrelatedness (and I would like to define uncorrelatedness by $(1)$).
What am I missing?
If we would indeed define mutual uncorrelatedness of rv's $X_1,\dots,X_n$ the way you propose then we will meet the unpleasant situation that "mutual uncorrelatedness" of $X,Y,Z$ will not imply pairwise uncorrelatedness.
This in contrast with independence.
This causes possibly more confusion than convenience.
I think there are not so much situations where this terminology would be convenient.
Further in each of these situations it is enough to say $(1)$ is satisfied (without using the proposed terminology).