Is a plastic drinking container without a handle properly called a CUP or a GLASS?

I’m a native speaker, but disagree with some people over the proper term for a plastic drinking container. For the record, I think it should be “cup.” However, I’ve been handed teacups by people after asking for a cup and had them insist that a cup must have a handle and that my “cup” should be called a “glass,” even though it’s made of plastic.