Thanks for your response Jonathan. "Oblique" anything does have great appeal to me as at least you don't have to introduce extra terminology. The "equilateral triangle as isosceles" is one I understand but then we still have the same issue as with quadrilaterals: what do we call (concisely) the non-equilateral, isosceles triangles? Seeing as though both Australia and the US have just released national curricula it'd be great if they could also specify what they mean by the terms they use, like this one from the US Grade 2 Common Core Standards:
"Recognize rectangles, rhombuses, squares and trapezoids as examples of quadrilaterals..."
While later on in Grade 3:
"Describe, analyze, compare and classify two-dimensional shapes by their properties and connect these properties to the classification of shapes into categories and subcategories (e.g., squares are "special rectangles" as well as "special rhombuses").
It's clear that there is a jumble of inclusive and exclusive definitions at work, made more difficult by my original point - there aren't enough names for the "endpoints".
An illuminating review of what definitions are actually used in textbooks is "The classification of quadrilaterals: a study of definition" by Zalman Usiskin and others. I got it just the other day and while it isn't ideal bed-time reading, it does reveal that there is much confusion between and within publications. I suspect not much will change unless some authority sets out some consistent definitions or at least guidelines for discussion with students.