I think that after I’d gone to Yadong Hospital excessively in the old days (heck, I still spend a lot of time there), they must have got tired of pronouncing my English name and just slapped a three-character Chinese name on me. And then I guess during one of my ARC renewals, I must have given the ARC folks that name, maybe from my National Health Insurance card.
But I don’t know; for all I know, maybe there is some requirement these days, like the one you mentioned, so maybe you shouldn’t use my experience as a guide.
Edited to add:
Here’s a 2016 thread that seems to be about that subject (I don’t know whether things have changed since then):
2011 thread, related subject:
Here’s a discussion from 2003:
A poll and discussion from 2020:
Another thread from 2003:
In a 2002 thread, one poster seems to express surprise that foreigners are allowed to have Chinese names on their ARCs:
A 2020 thread on chops (name stamps), and whether one needs to have a chop: