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So I agree with all that: it is shameful that these groups are not being targetted and outed. My point is that treating nazis as separate from white supremacy groups does not help in targetting these groups. What does it matter if they have sperate thinktanks? It's not like these same people would be in either group if the other were not an option. It's not like these people would not be radicalized if they just didn't hear a Hitler quote or like the SS uniforms. The causes of the two groups is the exact same thing. I feel like you are saying not differentiating doesn't help in bringing back white supremacists to a non-radical ideology; which I agree with. If I say to a white nationalist you're are a nazi, I'm not going to change their mind about anything and they will just think I don't know what a nazi is because that is those other guys in that other discord channel. I don't care about people who are already lost - or I should say I don't think that what we call them has anything to do with helping them or entrenching them further.
My point is that by differentiating them we tend to put a hierarchy of badness where we say Nazis are terrible, they want to take down the government and install a dictator and kill all non-whites while white supremacists, really they just want to go back to the good old 1950's where you could lynch a boy for getting out of line (and ignore that really they want to take down the government and install a dictator and kill all non-whites as well). In my view, they are more easily allowed to infiltrate institutions by looking like the latter which we fined more acceptable because of its less alien to US culture. But at the end of the day, they amount to the same thing. It also should stop people accepting moderate white nationalist views if they understand the consequence of that is what we saw in 30's Germany. So sure not differentiating them might not change any of these peoples minds but it might make it look like a less acceptable thing to get into in the first place. Prevention rather than treatment.
Also, I completely disagree that it is mine and other peoples generalization which is what lets these groups thrive. You are right that they are slimy and will try to pin you down on definitions and they do use confusion but that's only because we play into their hands by saying a nazi has to quote Hitler etc. and not saying a Nazi is someone who wants to create a white ethnic state through violence and ultra-nationalism. And - and this is an important point to my perspective - they would do this anyway; it doesn't matter what we call them they are slime and will try to hide in misinformation and confusion. However, the more people who will say a white nationalist is a nazi, the less wiggle room they will have and the more it represents their ideology in terms of practical outcomes instead of just more jobs for white people.