You would talk of a "random variable" (loosely, a quantity that exhibits random behavior, although there is a technical definition that is more precise and has a different flavor), not a "stochastic variable".
Sometimes, you would talk about a "stochastic matrix" (which is actually deterministic, an example of a namespace clash), because a "random matrix" would indeed be random be filled with "random variables".
But, you would often talk about a "stochastic process" like Brownian motion. Although you can also say "random process", and that's the same thing.
You would not talk about a "random distribution", that sounds like you mean "arbitrary distribution" (as in, "I picked out some random socks today"). But you don't mean arbitrary, you mean a "statistical distribution" or "probability distribution".
To get all this right, you have to live in that world, you can't just visit. If an outsider could just speak the language using clear rules, there would be no point in all those advanced classes, right? ;-)
By saying "this is a random sequence," you enable a statistical view of the problem. Even for purely deterministic sequences, this can be useful.
Last question, something I've never been perfectly clear on, what is the difference between "random" and "stochastic"? My understanding is that a stochastic model is a model featuring stochastic variables, which are variables that draw from some random distribution. Is the terminology I am using correct?