Incorrect.
EA found a way (debatable as they gave up when lawyers where brought in) around Steams already established rule that ALL PARTS of a game (DLC included) was on STEAM. so end users had access to all parts of the game in one spot. Valve did rewrite the terms after they tried this (so you are correct in that part) to make them crystal clear, and leave no loopholes to use. When this all went to lawyers EA, not Valve, pulled games. In the legal maneuvering Valve was found to be in the right, (they only reason you see EA games still on steam that where there before this, they would like to have pulled them all). Lets not forget EA had a digital download service BEFORE valve did. They did nothing with it, prices where more then boxed copies in stores. It was in place for those who wanted copies hours from release. They did nothing to promote it or make it worthwhile. Steam came along and cornered the market. It wasn't so much STEAM wanting a piece of everything, but they do want everything available for all users on their service if they support the main game (making money on it is a byproduct for sure). But more EA wanted the whole PIE, which after STEAM made it well known and used they felt having 2-4 big names game games them the right to do. Its simple, go look at sales before and after STEAM for EA, they are less.
VALVE is in it to make money no doubt about it, but to make them the BAD guy in the EA thing is incorrect. All the info is out there, simple to read (maybe a bit too legal to understand) this was all debated to death when it happened.