Does what the trailer says about Atari defining what a video game is stand up? I think there were other influential game makers and home systems around too, right?
Yes in a way. It did for gaming what NES did for gaming. You are right about other consoles which were superior - Intellivision and especially, Colecovision, but those came out in 1980 and 1982. Atari came out in 1977 and it was a catridge based system that did not rely on TV overlays, or built in games only, and had various controllers to plug in, etc, and tried to do arcade conversions. Prior cart based system was very simplistic.
Plus, while it had a TON of shovelware, it introduced to home video games some real good games that went away from arcade style, such as Pit Fall and Adventure, and hell, Indiana Jones was a point and click adventure game that needed both controllers to play (inventory management).
And it did have some cool original shooters, like Yars and River Raid, etc.
It got outclassed by Intelivision and Colecovision pretty hard, but Atari had enough of a head start and cultural cache for breaking ground. Atari's problem - well one of - was that technology was moving so rapidly compared to the present and yet they rode the 2600 way past the due date, and they had a successor in 1982 (5200) which they did not support very well and it was still not as good as Coleco and came late to the party. On top of that, no backwards compatibility and no showcase games, as most were "enhanced" ports of 2600 games.
By the time they had the 7800 in 1986, it was already outclassed by the Sega Master System, the NES, and the PC Engine which released just one year later in 1987.