Many of those words aren't actually limited to those use cases, but they are used there because they have very specific meanings. A plaintiff, for example, is the person lodging a complaint. Doesn't have to be a legal complaint, but in legal terms it makes it very clear who we're talking about.
Others are just expressions. A snap election is just when you call an election earlier than the scheduled one. It's essentially a nickname for something more complicated. Same goes with casting a ballot. It just means to toss your vote in for the count. It's just the expression that stuck because it sounds fancier than just saying "voting."