Pom or Pommy is an Australian derogatory slang for an English person. I believe it originates from P.O.H.M. Stamped on the prisoners uniforms. That stood for Prisoner of His Majesty and is meant as a an insult to Pom’s regarding their lack of true freedom.
Naturalman75 can correct me if I have it wrong since I’m not a native Aussie.
As one called a Pommy Bastard when I came to Australia in the late sixties there are several versions of the story. I think the one you gave holds up as well as any. The thing about Aussies, especially those of older generations is that they only really abuse their friends (mates). It may be derogatory in a sense but it is meant to people to not be a “whingeing pom” but to join in with the rest. Same with Italians, Greeks and other Europeans who came here after the war were called Wogs, Spicks and Dagos. Some may have been serious about it when it first happened but they became familiar friendly jibes we all called each other. We all had to become Aussies. Similarly Aussies were very often called Skips (as in Skippy the bush Kangaroo) we are a strange lot but I love this country with all my heart and especially its peculiarities!
The origin of Pom is debated - the abbreviation argument is possible (both POME and POHM) but it never seems to have been a formal abbreviation. Another suggestion is it derived from rhyming slang - Pomegranate for Immigrant, although that has never seemed like a particularly good rhyme to me :)