Balrog doesn’t originate from the Udun nearby Morannon. When Gandalf calls it ”The Flame of Udun”, he means Utumno, the First Realm of Morgoth. Udun is a Sindarin form of that name. It is very likely that Sauron had named the later Udun after Utumno.
Annúminas was the first capital of Arnor though, but it fell into ruin due to losses in the war against Sauron. Afterwards the capital was moved to Fornost.
You're totally right about the Durin's Bane one, changed the question. For Udun, looking at the enlarged map of the southeast from my edition of Return of the King it seems south is more appropriate, especially since some of it peeks to the west of the Morannon. Thanks again for the corrections!
Annúminas was the first capital of Arnor though, but it fell into ruin due to losses in the war against Sauron. Afterwards the capital was moved to Fornost.
Two comments:
- Udûn is southEAST of the Morannon
- "Balrog" is not really a name for Durin's Bane, but the type of creature he is.