I have sometimes heard of people who distinguish between Monte Carlo algorithms and Las Vegas algorithms. Unlike a Monte Carlo algorithm—which will always terminate, but has a chance of giving wildly inaccurate results—a Las Vegas algorithm has a chance of running for an arbitrarily long time, but will always give accurate results. I suspect that most people, even algorithms researchers, don't make the distinction often, since (as you note) most people use "Monte Carlo" and "random" interchangeably. (Wikipedia says that there are also Atlantic City algorithms, but I had never heard of that term until now.)