"We" is such an imprecise word for a pool of people. I believe Chinese has two flavors, "zanmen" including the listener too, and "women" excluding the listener. Obviously "we" did not elect Trump, only "a majority of the US voters who voted", and even the others may sadly use "we" though they didn't, because they are members of the political body that did. Just like the "they" of Israel that harass Palestinians and throw up West Bank settlements do not reflect all of Israel, and the average Soviet citizen did not reflect the behavior of the Soviet government.
We isn't an imprecise word at all, it's very precise in it's definition.
I can honestly not come up with a single example of the distinction between 'zanmen and women' being useful besides this specific case where you really want to be able to say in 1 sentence that you identify as the same group as someone else, but that that group is subdivided into 2 groups, and you're talking about the sub-group that you're specifically not a part of.