The individual people who did that and were in favor of it are/were nazis. Labeling an organization as nazi/not nazi is different than labeling an individual.
It’s impossible to know a person’s mind. We can’t possibly hope to define people based on their beliefs, because they are impossible to verify. We can only judge and identify people based on their actions alone.
If an individual person inwardly truly believes that we should kill all jews, but their entire life they never say so. They never do anything to further that goal. Somehow they manage to continue holding that belief while also acting contrary to it. That person is not a nazi.
Opposite is the same. A person who deep down inside believes all people are created equal and murder is wrong. Yet during their life they take actions to further the cause of white supremacy. That person is a white supremacist. A person who votes Republican even though in their mind they hold no racist beliefs, doesn’t matter. They’re a racist because they have used their power to further the cause of racism, among other things.
Actions define who or what you are. What you are can change over time. This rule applies to individuals, if you want to label an entire organization that is more difficult.