Usually, whether an assessment is a test or quiz will depend on what part it plays in the overall grade. A test is a large assessment that usually covers several complex concepts, and success or failure will have a large impact on grades. A quiz is a much smaller grade that often focuses on a single idea, reading section, or vocabulary list, and a single quiz will have minimal impact on the overall grade.
It's possible to have a short quiz and a short test. This really isn't a grammar or usage question. It depends on the assessment that's described,
“A quiz” in England is generally a light-hearted affair, as opposed to a test or exam. Compare ‘a television quiz show’ or Trivial Pursuit™ which is a quiz, but in the form of a board game. I doubt that ‘quiz’ would be used scholastically, in England, except perhaps to refer to an end-of-term knockabout. (On the other hand, to be ‘quizzed by police’ is generally not a light-hearted affair: quite the contrary, in fact!)