I’ve never heard the term wafer cone. Sugar cones are, I suppose, the proper cones but I’ve never liked them(they taste like fortune cookies to me) and if a cake cone isn’t available I’ll ask for a cup. I’ve never had a waffle cone.
Slight hijack; I’ve mentioned this here before I think - recently a local radio host said adults look ridiculous eating ice cream cones and should always use a cup. I was shocked at the number of callers who agreed? WTF?