Is this a terrorist organization? Why don’t extinctionists promote murder?

No. Extinctionism is against violence. Going into a forest and ending a deer means they just reproduce and create one more deer because population of non human animals depends on amount of resources. Ending any individual also causes suffering and pain to this individual the moments before death, and to those close to them because of emotional trauma. So this increases suffering. Extinction, on the other hand, ends the cycle of suffering.