Twitch/Internet culture treats women badly.

Victim blaming disgusts me. If someone hurts you it's the fault of the person who did the harm, not the person who was at the end of it.

Is it weird to say I agree with both of you? To an extent?

Yes, trolls and those who use words or other methods of harassment, do it to incite behavior for their own kicks and giggles. To recognize this behavior at least lets you know what you're dealing with.
However, it's not cool to "troll" this way. It's harassment and that's not ok. Ignoring it doesn't always mean it goes away. Standing up together against this kind of harassment is a great way to let others know what our social norms are.

State a way to troll that is cool

Haha, good call. I left that open in case there's ever a way people can incite a response without being rude or mean, but it's just how they like to mess with people. I don't like using always/never as I like to think there's a possibility out there. But yah, doubt there's many ways to "troll" and be cool.

Correct someone and be as smarmy and condescending as possible while doing it. Does that count?

Wouldn't call that trolling

While yeah, I do think it's wrong to get baited into fights with trolls and give them a response, I still think just saying "Oh, it's the internet, people are just sh*tty" is an awful attitude that solves nothing

Nope. I do agree with him that it should be fought. I disagree that the victim is helpless tot watch. The first fight tot win is to recognize a troll.

I've had tot display immense willpower in the past and found that if you know the other only says it to flamebait with no sincerity behind it, its simply not effective

Intentionally disregarding personal faults disgusts me.

The only way to improve is to understand faults and rectify them. It's not about placing blame, it's about doing what you can to make your situation better.

The best way to get trolls to stop is by not feeding them. Anger feeds trolls, bans feed trolls, call outs feed trolls. If a person isn't trolling a single "yo that's not cool" should be enough. Any more than that is more than just wasted effort, it's an active detriment.