Are there any specific reasons why Carbon based compounds are termed as organic?

It seems arbitrary, because it is! The general way to classify organic and inorganic compounds are by the definitions themselves. If the compound has carbon and hydrogen and occur naturally, it is an organic compound. The other compounds that are synthesized or man-made, it is called as inorganic compounds.

Diamond is made of Carbon, yet it’s not classified as an ‘organic compound’.

Diamond is an allotrope. Means, it is just Carbon existing in another form. So it is basically just Carbon, a substance.

There is not a clear definition available on either cases. As it is mentioned above, the naturally occurring compounds in living organisms are usually called as Organic compounds. This is vague in many cases, but almost true to cover the definition of ‘organic’.

But Glynn’s definition of ‘inorganic compound’ is too ambiguous. There are non-synthesized compounds inside inorganic classes too. A simple example is Water (H₂O). There is no Carbon in it. You can even synthesize an organic compound from inorganic reactants.