Thanks for replying Walter. The diagram I linked to in my original post comes from a book that also has the quadrilaterals set out according to symmetry - a way of organising them I'd never thought of until I'd seen your work. I'd forgotten about your label for the non-rectangular isosceles trapezoid - "butterfly" - so I'll make a note as it's a useful label that nicely links to the reflective symmetry of the shape.
While reasoning and proofs are an ultimate goal, my concern is mostly at the very basic level. There are many problems with labelling shapes too early as it can lead to poor understanding. For example, attaching the label "rectangle" to only shapes that are oblongs often results in students being unable to identify squares as rectangles too. This occurs from age 3 until even age 12 in some cases (sometimes beyond unfortunately!). So it seems important that we should use labels that always make it clear when we are talking about general cases ("rectangle") and when we are talking about specific cases ("oblong" and "square"). Imagine if we showed kids pictures of Saint Bernards exclusively with the label "dog" for the first 7 years of their lives. Then trying to convince them that a Chihuahua is a dog too will be a difficult process! (I'm still not convinced either, but for different reasons.)
For better or worse, the "end-points" are where kids begin learning about shapes. The standard set of pattern blocks is a classic example of where the problems creep in. There's nothing wrong with the shapes, just the labels. If we accept for a moment that we can use 2D names for them, we have an orange square, a green triangle, a yellow hexagon, and then... well, this is where the problem occurs. The blue shape is a rhombus, but it is a non-square rhombus, just as the pale rhombus is too. But if we attach the label "rhombus" only to non-square rhombuses, should we wonder that kids have difficulty later with understanding that a square is also a rhombus? Similarly with the red shape, usually called the trapezoid. Again, it is a special trapezoid (non-rectangular) but giving it the class name of "trapezoid" causes confusion all through childhood and into adulthood as people struggle to see how oblongs and squares could possibly be included in the class of trapezoids.
Whether the chart I linked to is used, or the chart you present with Lily Moshe is used, we still have "name gaps". There is no exclusive name for a kite that is not a rhombus. There is no exclusive name for an isosceles trapezoid that is not a rectangle. There is no exclusive name for a rhombus that is not a square. Putting aside higher-order thinking about the relationships between shapes, when I have the humble task of asking a young child to draw a non-square rhombus, it would be really handy to have a label for it apart from "non-square rhombus".
P.S. I came across the Shreddies ad just a few days ago - very cute, but I agree, it does reinforce the idea that a diamond is based as much on orientation as on shape.