Logically there are two ways one could be a 'white supremacist', and they are related.
The first would be to believe that whites are the superior race, better than all the other races. Even well known alt-right and racialists like Richard Spencer don't agree with that. He's been asked it, and said "all races have their strengths and weaknesses". You can't look at the starting line of any Olympic sprint race and claim that whites are superior in all things. (But you also can't look at it and claim that genetics has no role in high performance in short distance track & field events).
The second, related idea, would be that because of innate superiority whites are destined, obligated or gifted by god to rule over other races, possibly including holding them as slaves.
If you don't believe in the first idea, the second one makes no sense.
Both these beliefs were common a couple centuries ago. There were all sorts of justifications of slavery put forward with appeals to the Bible. But eventually they were overturned, the strongest advocates for abolition were devout Protestants.
And today nobody believes those things. You can scour the net and not find anyone arguing for either the generic superiority of whites, or the idea that whites should therefore rule over other races.
So, it's a completely bogus term. There literally are no sefl-avowed "white supremacists". They only exist in the minds of leftists, and when I hear anyone say that term I discount what ever follows, because they are completely ignorant and haven't even bothered to learn the basics of the thing they are pretending to know about.
“White supremacist” is a term the Left has taken to using because labeling everyone they disagree with on policy a “racist” isn’t working well any more. People say FU, blow it off, and demand a debate on policy merits the Left can’t handle.