Do you have James and Elizabeths actual marriage record to confirm what his actual middle name was (not a transcription)? It (Auarther) could be a transcription error or could be his actual name. For example, there are people around with the last name Anarther but not Auarther.
If you dont have the actual marriage record, then that would be your first place to start, confirming whether he was this Anarther/Auarther or was he Arthur with an odd loop of copperplate writing that was transcribed wrong.
Middle names back then were also here or there especially immigrants. For example, I have a person in my tree that immigrated to Australia with no middle names, married without any mention of them but when his death was registered in 1900, he ended up with two middle names.
As for the newspaper article you are wrong in your thinking. The birth notice says a son was born to James Anarther Williams on 1 May but the actual date of the paper it was entered in on was 16 Jul 1891. A full 2 months after his birth. Highly doubtful to be an error or malicious especially given that in the same column at the start it clearly states:"Announcements under this heading must be authenticated by the name and address of the sender..."
As I mentioned, this also could be a different person with a similar middle name, so it is hard to know exactly especially given the child's name isn't given. So, need to confirm that middle name before anything else.
Have you checked interstate for James' birth? He could, as you say, also be a new immigrant which can prove very difficult to trace especially in Australia back then when fact checking wasn't possible.
