You're pretty much right. BT is slightly different in the sense that it seems a planet gets a name which then also becomes the sun's name, but then the official indicator is usually adjusted based on that.
In the case of the Terran system, it appears as 'Terra' on the map, not Sol. So the fourth planet in the Terra system would be Terra iV.
Likely a split between how astrologers would do it, and how people in early space flight would handle it, so the colloquialism won out in the end.
Where do you need to go sir?
- New Earth.
Oh, I do believe you mean Tau Ceti IV, don't you, sir?
- Yeah, you know what I mean, you pedantic bureaucrat. How about I ask your manager.