There's only one type of toxic player in the League of Legends community.
If someone is toxic towards others, such as blaming them, telling them that they're useless, or pretty much anything that the person would never say straight to their parents' face, that is toxic behavior.
It's easy to tell if someone is toxic, or will post something toxic. The above is the indicator: If you can't say it to your parents' face, or your significant other's face, you probably shouldn't say it at all. There's times to trash talk, and there's times to not. The times to trash talk and make your teammates and opponents feel bad is NEVER. The times you should not trash talk is all the time. Easy as that.
Scream at your screen, not at your teammate. Vent your rage with a stress toy, not crucifying your teammate or another person. It is NEVER cool to make someone else feel like crap just because you feel like crap, and that's something I don't think toxic players in this game understand.
For reinforcement to that claim, here's that age-old analogy that I'm pretty sure everyone here has heard of at least: "Treat others how you would want to be treated". If you treat someone nicely, they'll be nice to you.
There is no two types of toxic player; there are two types of players though: The good player, and the toxic player. People of this community should strive to be the former, and never the latter.