To add to the confusion, and to emphasise the point about making
clear, British and American English differ here. If three
proposals (not the most common case, I agree, but it can happen)
have 45%, 35% and 20% of the votes, the first of these has a
majority, sometimes emphasised as simple majority, in British
English. (We do not - to our loss - use the word plurality.
Just 51% is given the strong term absolute majority.) I haven't
checked the context here, but saying not just a simple majority
might suggest to a British English user that 51% is enough.
Let's avoid any dependency on either idioms. Given that the bulk
of the intended audience might not be aware of either specific idiom
but have english as their second of nth language, it is probably
better to explain what is meant.
jaap