And, after I hit my head a little on my table, I understood.
For everyone who comes for me:
I had to change a couple of things in the application descriptor file.
First I had to change the namespace to indicate in AIR 1.5.3 instead of 1.5
<application xmlns="http://ns.adobe.com/air/application/1.5.3">
And then I had to manually specify our old publisher ID in the descriptor, as well as
<id>OurAppID</id>
<publisherID>OurOldPublisherID</publisherID>
Now it works as expected, and installs a new version instead of the old version, and instead just launches this error.