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: 
The IUPAC definition of element is as follows:

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> A pure chemical substance composed of atoms with the same number of protons in the atomic nucleus. [IUPAC Goldbook definition of element](https://goldbook.iupac.org/terms/view/C01022)
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This seems to allow diamond to be called an element. But it's not a good idea, scientifically or linguistically, since it doesn't make sense.

Carbon is an element. Diamond is a *form* that the element carbon can take.

Let me give you a car analogy:
Toyota is a brand of car. The Camry is one example of a type of Toyota car. I.e., it is one form a Toyota can take. That does not make "Camry" a brand of car, nor would it make sense to call a Camry a car brand.

In summary, referring to diamond as an element, rather than a form an element can take, is like referring to Camry as a car brand, rather than as one of the brand's models.
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.