I know there's a page on bulbapedia that has all the types of trainer classes but what I'm wondering is there a list or database of all their individual names. Like all the bug catchers names, all youngsters names, all ace trainers names etc.
There is no one page with a list of every trainer's class and name, but there are a collection of pages to visit. Go to Bulbapedia's Trainer Class page and there is a list of each trainer class fittingly.
Next, On each trainer category page (e.g. Bug Catcher, Ace Trainer, etc.) There is a section called Trainer List. Under this is all their names and games they are in.