When Angelus has his soul back he goes by the shortened form Angel.
Is there an established in universe reason why he doesn't use his real name Liam?
While this is never definitively stated within either of the two main shows (possibly within a comic or commentary), it is my belief that Angel effectively considers "Liam" to be dead, and that identity completely wiped out.
From the episode Amends (In this scene, the First Evil has taken the guise of Jenny Calendar):
Angel: A demon isn't a man. I was a man once.
Jenny: (derisively) Oh, yes, and what a man you were.
Angel is forced to flash back to the days before he became a vampire, in a bar, singing and drinking, pulling at a woman's dress before passing out and falling to the floor. The flashback is over, and Margaret is standing before him again.
Margaret: A drunken, whoring layabout, and a terrible disappointment to your parents.
Angel: I was young. I never had a chance to...
Margaret: To die of syphilis? You were a worthless being before you were ever a monster.
It is theorized that Vampire-Liam took the name "Angelus" from his sister's belief that he was an Angel, but he has been described as "having the face of an angel". After he has his soul returned, he is no longer the person that Liam was, nor is he the murdering demon that Angelus was. He is now a third, composite personality/being, and taking a third name is a symbolic gesture of that. Keeping it close to "Angelus" fits with his mission of redemption and acceptance of his past.
Out-of-universe, there is a certain amount of logic to the shortening of "Angelus" to "Angel" as one might see (as many vampires do) Angelus being weakened into a lesser being.
When Giles and Buffy researches Angel after Buffy discovering he's a vampire, he is refered to as "Angelus; the one with the angelic face" in a book. My theory is that Liam takes the name Angelus - or perhaps more probably, that his sire Darla bestowes it upon him - after he becomes a Vampire. When he gets his soul back and remember all the horrible things he's done (as Angelus), it's not that strange that he "creates" "Angel" as a way to keep his sould and un-soluled selfs apart.