Not precisely on point, but the concept of a "heckler's veto" is the name of a voting system that allows one person's disagreement or complaint even in a large group to override decisions of the majority.
For example, when I was a freshman in college, in a co-ed dorm, each floor of the dorm had a vote on whether or not the bathrooms would be co-ed on that floor. The decision was not made by majority vote, but by consensus. Anyone living on the floor could employ a heckler's veto.
In the U.S. Senate, many actions are taken based upon "unanimous consent" to which any Senator may object and force the usual process that is slower and more cumbersome to be used instead.
Part of the difficulty with the original question is that the example given doesn't really match the question.
In the examples given, a decision is made by a majority vote of an HOA board or a city council, and the outcome is a function of a perception of what the public wants as well as the board or council member's own decisions. Responsiveness to vocal minorities is different from decision making by vocal minorities. And as such doesn't really fit with other "-cracy" terms which usual refer to who actually makes decisions, not to who lobbies and influences decisions. You could still say that it is de facto rule by a vocal minority, but that wouldn't be the conventional way to describe it.
Often politicians would accuse officials of "ignoring the silent majority" whom their opponents would claim to represent, or of being too solicitous of "special interests", or of being "too politically correct" (often abbreviated "too PC").