I tried asking around the Malta Study Circle and on another forum as well, but I got no answers. I'll try contacting the Malta Philatelic Society and the Gozo Philatelic Society as well.
I also went to the GPO to ask if they have any information, but no one was able to help. In 2012 I obtained a money order from the post office, but it had no stamp, just a regular Valletta postmark. Today MaltaPost still offer a money order service but it's simply a bank transfer, and printed forms like the one above are no longer in use.
https://www.maltapost.com/local-for...money-ordersThe text in the section where the stamp is affixed reads
Timbru tal-hrug, which is Maltese for "postmark of issue". This refers to the cancellation and not the stamp.
The idea that the "stamp" might be a counterfoil seems to make sense.