Okay. Here's how the System works:
The publisher is at the top of the chain. You'll note I have a crown over the publisher. That's because I'm obnoxious. The publisher wants a game made. The publisher either creates the game in-house, or contracts a developer to make the game for them. Let's talk about how the outside developer is handled.
Usually, the outside developer is pretty much told what to make. Something based on a cool movie or TV license, or a sequel / spin-off to a game another studio originally made (after the publisher has happily said to the original developer studio, "Go jump in a lake, we own the property, we don't need you, so NEENER!" or said studio has vowed "We'll never work with you again, you jerks! For definitions of 'never' that include this product cycle!"). Occasionally, the outside developer might have some cool proof-of-concept demo that the publisher is willing to go with, so long as the developer makes all these changes to it (usually converting it into something that is based on a cool movie or TV license, or a sequel / spin-off to a game another studio originally made… okay, you get the idea).
The publisher "funds" the development of said game. By "fund," we really mean, "loans money to the studio for." Because… really… funding is an interest-free advance against future royalties earned by the game. Which, according to modern accounting practice in the games biz, is actually just a myth and never really happens, so the loan almost never gets repaid, so the publishers can act all magnanimous about it. But hey - it's their investment money, and so they get to call the terms when they want to it back two different ways, that's their call.