defining properly the notion of "class (or type) of quadrilaterals"?

Question

Many people and teachers have their own idea about what are "the" classes of quadrilaterals, and they are not the same for everyone! Is it possible to explain what can be called a class of quadrilaterals? So far, I have not found anything else than lists of classes - some of them being "special" (?), "important" (??), "particular" (???). Thanks in advance for your comments.