To be honest, a diamond is not technically a mathematical shape whereas a rhombus is. When someone says the word rhombus, you know they are referring to a quadrilateral that has all four sides the same length; the opposite sides are parallel, and the opposite angles are equal. (Mathematical Warning: A rhombus is not thinner than a diamond, AND the plural form, rhombi, is not a dance performed on the program Dancing With the Stars.) 
But what comes to mind when you hear the word diamond? If you are a woman, you might envision a large sparkling gem setting on the ring finger of your left hand. If you are a guy, you might think of a baseball infield. (The distance between each base is the same, making the shape a diamond.) If you play cards, the word might bring to mind a suit of playing cards, OR you might recall a line in the song, Twinkle, Twinkle, Little Star. Calling a rhombus a diamond is similar to calling a child a "kid" (could be a baby goat), or a home your "pad" (might be a notebook). The first is an accurate term, the second one is not.
So how does this affect you as a teacher? It doesn't, unless rhombus is on a local benchmark or state test. But if you are an elementary grade teacher, please use the correct mathematical language because a middle school math teacher will thank you; a high school geometry teacher will sing your praises, (see song below) and a college math teacher, like me, will absolutely love you for it!
Rhombus, Rhombus, Rhombus
(sung to the "Conga" tune)
(The song where everyone is in a line with their hands on each other's shoulders)
(The song where everyone is in a line with their hands on each other's shoulders)
Rhombus, rhombus, rhombus;
Rhombus, rhombus, rhombus
Once it was diamond;
Now it's called a rhombus.
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