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From my understanding Capcom and Konami were close to each other in terms of size. Premier most certainly can mean multiple by the way, there's multiple premier restauarants, premier game developers, premier engines(Cry Engine, Frostbite 3.0, Unreal Engine 4). Yes Premier can technically mean "best", but it's usually referring to a standard higher than other companies in an industry. In terms of industry, that is how the term is almost always used.
Capcom had : Resident Evil, Street Fighter, Ghost N' Goblins, Final Fight, Mega Man
Konami had : Adventure Island, Bomberman, Castlevania, Contra, Metal Gear, Silent Hill, Suikoden, Zone of the Enders
I think the reason people don't remember Konami as a "premier" Japanese publisher is because eventualy Capcom got franchises like Ace Attorney, Devil May Cry, Onimusha, Monster Hunter, and Konami franchises just haven't really made the switch to modern-ness. Castlevania and Metal Gear and Silent Hill are their only franchises that have had popular modern interpretations and even then Metal Gear has stopped production for the most part. We're lucky we got Bomberman. But still yes, Konami used to be one of the premier Japanese publishers back in the day.
Namco Bandai, Sega, and Square enix(though they're more so devs) are big too.








