So Mian an Ahn are names that were both invented by Heagu?

Mian means sorry in korean, thats why Haegu calls her like that

What about Ahn? The first time it was brought up I thought that was her real name

Ahn, I'm not sure, but I believe it's a smaller way of saying Mian, because the pronunciation of Ahn is the same as "an" in Mian.

Yes, Ahn is a nickname for Mian. Like mentioned above, it's just the second part of Mian.