Nobody is saying that all game developers are software engineers, or that all software engineers are game developers.
But software engineers who work on video games absolutely are game developers.
You're confusing "designer" and "developer."
There are different aspects of game development. Being a game developer literally just means you're one of the people that "makes video games," as opposed to people involved in publishing, testing, marketing, etc.
That includes directors, designers, concept and technical artists, sound engineers, and, yes, software engineers (programmers).
The software engineers are the programmers who build the underlying systems that make the game function, or the tools that help the designers do their jobs.
Basically, they do the under-the-hood work to make the game do what the designers want it to do, or they make tools that enable the designers to do it themselves.
As far as I know, Omar was mainly a gameplay programmer, though he also worked on programming the servers at different points. Most recently he was in charge of getting WoW Classic to run in the modern WoW game client.