Moon Orbits are assuming Orbits of their own.

I've been building systems since the latest patch and when putting moons in orbit of a planet everything runs fine in the System Editor, but when I Start a game the moons assume independent orbits around the Sun.

Burrito

Did you marked them as starting planets?

I suppose you still unable start on moon so all starting planets will rotate around the sun: PA#2835

I did indeed.
That's a shame but thank you for the pointer