Were the ruling elite of Tiamat also Anunnaki? I know the Anunnaki manned the Death Star, but I'm not sure if they were also the ruling elite?✶
No. The Mars natives had the same origins as those from Tiamat, with slightly different evolved bodies.
How they look now: The Martian natives (some survived underground and evolved) have an enlarged skull that looks as if a ball has been placed on the back of the skull. The beings from Tiamat (some now located in the moon) have a smoothly protruding (to the back) enlarged skull — like Nefertiti's headdress in the famous bust of her.
Would the names of the races of these planets be familiar to us?
The Martians were called Maldekians, in their own language, Maldekli. (Mars was called Maldek at a later period, after it became a desert.) The race on the planet Tiamat were called the Tiamaya (meaning 'the Children of the Eternal Mother'), in the ancient Mayan language.
The ruling elite from both planets were called the Ufizi, meaning 'overseers'. The Nibiruan soldiers stationed on Mars to protect the piece of Tiamat that became the Earth were called the Annunaki.
There were some of the more ancient Builder Race living in underground cities on Tiamat that were also destroyed when the planet was annihilated.
You had also given their sanskrit names?
The battle of Tiamat and Mars is described in the Hindu scripture of the Mahabharata. In there the inhabitants of Tiamat are called the Vrishnis. The inhabitants of Mars were called the Andhakas.