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.
No. Diamond is not an element.

It is a name for a gemstone, a particular occurence of an allotropic form of carbon that may or may not contain other elements in addition to carbon.

And while "pure" diamond is an allotropic form of the element carbon, diamonds that are actually pure carbon are usually synthetically made, as most naturally occuring diamonds have impurities/inclusions in the matrix.