Richard:
I am not sure if I am being clear or complete. There is also history behind all this. I leave the whole color management on the Photoshop side. That is also the recommendation from all of the third party paper people who provided the profiles. So the icc profile is selected together with the "Photoshop Managed Color" option in Photoshop Print Settings, rendering intent etc. So if I was going to print on HP Adv Glossy, I would choose color profile named as such in Photoshop. Now there was a bug in the driver (there still is after 10 years, shows how careful HP is) that when you changed one thing, let's say the Paper Source or Paper Size, it would randomly change something else like Application management would change to AdobeRGB. At that point, if I had HP Adv Glossy chosen in the Paper Type, it would automatically apply the profile again even though the signal from Photoshop would be hey, I am doing the color management. I thought I was going crazy trying to figure out what was going on. Finally I decided to create these generic paper types without attached profiles so even if the Color Management changed, it would have no profile so it will print as it came from Photoshop. As far as I know, when you are using Application Management, the purpose of the paper type is only to use the correct black ink. I am fairly certain, as paranoid as I am, I would have done the experiments to compare the outputs from choosing paper types of HP Adv Glossy vs my home-brewed "Generic Gloss." That's going back many years though.
:Niranjan