They only used socialists in their party name because it was very popular at the time
They hated the working class , peasants and unions
Their pact with giant corporations and industries was only to benefit the elites of the party
Hitler hated communist and many Jews were also far leftist in Europe and communism members
Hey there. I'm an American historian and former history and US Government teacher with an MA in American history. This is my field.
You are correct. Socialism is all about the promise of equality for everyone, and the guarantee that every person, no matter who they are or what they do, deserves equal benefits of society. (Think about it: every socialist you've ever heard of has at least made this promise.) Obviously, promising this to the masses of poor people at the "bottom" of society can make someone very popular, very quickly.
Mussolini started the Fascist party (capital 'F') with the idea that the Italian people were betrayed and demanded more from society; in other words, nationalism. The last thing nationalism wants is equality; they want to set themselves higher. The term 'national socialist' popped up for basically marketing and branding reasons, selling the idea that everyone
in our nation deserves equal benefits, but not all of those others.
Hitler adopted fascism (small 'f', the general concept named for the party) to German culture and called his party the 'National Socialists,' for which the abbreviation of the German term is 'Nazi,' and there they are. They
hated the Communists and their lack of social hierarchy, push of labor unions, and insistence that the Nazis shouldn't own everything because no one should own everything. Hitler and pals couldn't wait to blame the Reichstag Fire on a Communist agitator, for example, executing the guy and then using the "attack" as an excuse to seize total power, which he never gave up.
Communism (which is based on socialism) and Fascism (including Nazis) are both authoritarian, which makes them both the opponents of democracy but they also are directly opposed to each other. If it helps to think of them in pop culture terms, you can think of them as Klingons and Romulans.
But yes. You are correct.