What Shape Is It?

I was with friends in Vermont this week and shared this picture with them:

To my surprise, they were not as irate as I was over this mislabeling. It did, however, lead to an interesting conversation:

Eric: Why are you so angry about this?

Me: It's not a rhombus!

Sara: This is a parallelogram.

Eric: What's the difference?

Sara: A rhombus is a slanted square.

Eric: Then what's a parallelogram?

Sara: A slanted rectangle.

Eric: So what's a diamond?

Sara: A turned rhombus.

While maybe not mathematically precise, these definitions worked for two friends that don't live in math education. It got me wondering what all the fuss is about classifying quadrilaterals. With quadrilaterals, the "non-math" default is to give the most specific name possible, like 'square'. Calling a square a quadrilateral loses so much important information. We never label something 'up' the hierarchy like this in context. 

In other contexts though, going up and down the hierarchy is common. If I ask students if they have pets, they might say they have a dog, then I might ask what type of dog and they may say a Corgi, but it's totally acceptable to say dog instead of Corgi. 

So what is the skill we're attempting to instill when we work on hierarchies? It seems my friends have a pretty good grasp on defining shapes, even if they might not see a square as a rectangle, rhombus, parallelogram, kite, trapezoid, and quadrilateral, they would certainly see their Black Lab as a dog, a mammal, a vertebrate, and an animal. Is it the math specific vocabulary that is the stumbling block?