Common question...
As someone who's been doing this a long time, and well, I guess part of the initial group who have caused this "problem" that people have run with....
Pro Department & Pro Issue generally refer to the same thing: A glove which was ordered thorugh a pro player, minor league player, independent player, college player, or via a connection at a glove company. PD/PI gloves have a separate "form" that an associate of rep will fill out. This is ENTIRELY DIFFERENT from the online builder. At least for Rawlings and Mizuno this is true. The part when people say you cant put a name on it is false. They are most likely saying that because they are trying to flip them. Im currently sitting in front of a bookcase of these gloves with embroidery, so I can attest to this.
PI/PD gloves do not have some special website, but have access above and beyond the online builder to the public for options including: Any pattern Rawlings has a die for, additional webs (think hinged single post), additional leather colors (this gap is almost gone now), additional lace and thread colors (again this almost the same to the public now) options like additional embroidery on the glove, things like "hand sewn welt," and a few others not coming to mind.
The number of people with "connections," seems to have increased over the years which is great, but at the same time it creates a new pool of people trying to rip each other off. I hope this helps.