Question: Is it scientifically correct to call Diamond an element? Carbon has a wide range of allotropes, so would it be correct to actually just call graphite, diamond etc an element. For that matter, could we call any molecule containing only one type of atom, an element?

Answer: 
Technically Yes, but Scientifically No.
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Diamond is made of pure Carbon, so yes. However, it is not on the periodic table nor is it classified as an element by **most** scientists, so on that case, no. It really depends what you think.
Diamond is a form of *pure carbon* (which is an element). Logic may lead you to believe that diamond is, therefore, an element. However, this is a bit circular because if you say that diamond is an element, then every allotrope of every element also falls under the same consideration. 

Also, in order for diamond to be a unique element, the molecules that make it up should have a different number of protons than any other element on the periodic table. Since a diamond is made of carbon, it is really just another form of carbon.