Atari Inc (1972-1984) is the "original" Atari. Nolan Bushnell founded it, and then he sold it to Warner, and then Warner milked the Atari 2600 until the industry crashed.
Warner broke Atari in two and sold the console division (as well as the main "Atari" name) to the Tramiel Family, but Warner kept the arcade division, which they renamed "Atari Games" because they considered what they kept to be a minority spinoff.
Atari Corporation (1984-1996) was the Tramiel-owned company that made the Jaguar and Lynx.
Atari Games (1984-2003) was the arcade division, but they made NES games under the name "Tengen" (and got sued by Nintendo) because they weren't allowed to use the "Atari" name in the console world. Time-Warner eventually sold this arcade division to Williams/Midway. They stopped using the "Atari Games" name, preferring Midway, and Warner Bros got it all back when they bought Williams/Midway.
After Atari Corporation (Tramiel family, makes of the Jaguar) went out of business, Hasbro bought it, and they later sold it to the French company Infogrames, who changed their name to Atari.
The "Atari USA" in this story is the US subsidiary of the French company. The American division of this company is apparently profitable and holding up their failing French parent, so the successful American division wants to declare bankruptcy (WTF?) in order to break away and get out from under their failing parent. I'm not sure how that works, and I have no idea what that means to the future of the Atari name (which is owned by the failing parent).