Which city is the "true" Diamond Capital of the world, and on what grounds can it claim that title?

Could you be talking about Antwerp, Flem-ish? It handles about 80% of the world's rough diamonds.

https://www.beldiamond.com/blogs/news/why-is-antwerp-the-diamond-capital-of-the-world

Being a member of the jewelry industry, I'd say Antwerp is the automatic response, but maybe this is a trick question.

Could also be Kimberley, South Africa, which was the founding seat of De Beers power. Though the Big Hole (the world's largest man-made hole) in Kimberly has been closed for nearly a century, the city still boasts an important museum which houses the largest uncut diamond in the world. Kimberley is where the Star of Africa was mined. If Antwerp is not the answer, I can't think of a better one than Kimberley.

When "googling" one also finds "Israel" as Diamond Capital, but no particular city is quoted. http://www.diamondscapital.com

Also the City of Amsterdam has been called so but had to drop that claim after WW2. https://www.forbes.com/sites/anthonydemarco/2019/01/12/where-to-discover-diamonds-in-amsterdam/?sh=476c1ea14de4

So I think that indeed Antwerp is correct as described at
http://www.sandersjewelers.net/articles/why-is-antwerp-the-diamond-capital-of-the-world

Kimberley must have been the capital of "uncut" diamond.

Indeed. We tend to refer to Antwerp as "the Diamond Cutting Capital of the World" which is somewhat different than Diamond Capital. Which is why I wondered if it were a trick question. That word "True" made me look for other answers.