We know that the Red Emperor is called Moonson, Lord of the Four Quarters, Staff and Pillar of God, Leader of the Egi, Shah of Shahs, and the Bright and Illustrious Emperor of Dara Happa. He rules the Red Goddess’ earthly interests – so Lunar Empire is absolutely appropriate (meaning literally where the Moon has rule).
So likely Lunar Empire is the most common term by both people within and without for the areas and peoples where the Red Emperor is the ruler. Remember that the Lunar Empire – like most pre-modern empires – is a household and dynastic thing and not really a state.
Some might deliberately use old titles as well, like Dara Happa or Carmania or Saird to speak of areas and peoples. But those don’t really exist as political entities any more (the exception is Dara Happa that exists as a religious entity and thus has some continued existence).