Who is in charge of defining what is hate speech and extremist behaviour?

The specific behavior that's being called out here - antisemitic, Nazi, sexuality- or gender-based hate, and white supremacist content - are pretty common definitions of hate speech and extremist behavior. Either way, he calls out Valve's own internally written content policies - which he states aren't being enforced - as the point of contention; he doesn't seem to be imposing outside views on them.

What if it were the people who don’t agree with your definitions is in charge of setting the definitions?

Then Steam becomes X or Truth Social, I guess? I think the chances of that happening are incredibly slim. A more likely negative outcome would be the terms being interpreted to broadly and positive speech being limited along with the negative, but to your point

Slippery slope.

Aren't you the one committing the slippery slope fallacy here? You're seemingly suggesting that a crackdown on hate speech will lead to or open the door to a bunch of negative outcomes.

Free speech is one of those things that is absolute. You are either for it or not, any encroachment is going to be the anti position. Obviously popular speech isn’t something that needs to be protected.

If you're defining 'free speech' as the ability to say whatever you want, wherever you want (including on private platforms), without facing consequences, then no, I don't support (your rigid definition of) free speech. I think that's a ridiculous definition to use, though, and I don't think it should be viewed as black or white. 'Free speech absolutism' is what leads to misinformation on the scale we're currently seeing (in the US). Furthermore, 'free speech' as outlined in the first amendment doesn't apply here at all.

Regardless, I don't like the idea of my kid (or any kids) being exposed to Nazi, white supremacist, or discriminatory rhetoric when he's on a gaming platform. Since that's specifically what Warner claims to be addressing here, I support calling it into question.

As Black Friday and the holiday buying season approaches, the American public should know that not only is Steam an unsafe place for teens and young adults to purchase and play online games, but also that, absent a change in Valve’s approach to user moderation and the type of behavior that it welcomes on its platform, Steam is playing a clear role in allowing harmful ideologies to spread and take root among the next generation.