I'm not quite sure what you're asking here, but there are several stories to tell behind that question.
First of all: What does the term "Nazi" mean?
Second of all: What did "national socialism" mean to national socialists?
So.
What does the term "Nazi" mean and how does it relate to national socialists?
Well, 'Nazi' is a German diminuitive name for the first name 'Ignatz'. I say "is", but the actuality of that is a bit iffy, as 1. calling somone 'Nazi' is not so nice and 2. no one is called 'Ignatz' these days. But okay, a hundred years ago there were a few Ignatzs, especially in the southern German cultural room, in Bavaria and Austria. An example? 'Nazi' was a derogatory nickname of Austro-Hungarian soldiers among their German comrades. Likewise, the Germans called British soldiers 'Tommys', for example. Another good example is a 1922 newspaper article by noted German author Kurt Tucholsky titled 'Die Nazis' in the newspaper 'Die Weltbühne' in which Tucholsky proceeds to make fun of Austrians. 'Die Weltbühne' is notable for the contributions of both Tucholsky and another eventual Nazi opponent, Carl von Ossietzky.
Well. One particularly weird Austrian during the early 1920 was one Adolf Hitler, whose German power base was furthermore located in Bavaria, where he also tried to coup the government in 1923. Connecting the derogatory nickname for Austrians "Nazi" with the "Nationalsozialismus" was just a matter of time - even though the initial preferred abbreviation was "Naso", which is more in line with how German abbreviations usually work. While the NSDAP was originally conflicted about using the term 'Nazi', they moved away from it by the 1920s due to its derogatory origins. The Nazis would never call themselves Nazis throughout their entire time in office 1933-1945.
The term found entry into other languages, particularly English, with the mass emigrations of Nazi opponents from Germany to the UK and the US. The term is also used in French, Polish and Dutch, among others.
What did "national socialism" mean to national socialists?
The term 'national socialism' is of course a spinoff of the term 'socialism'. The oldest usage of the term we have stumbled across is all the way back in 1887. Cornelia Schmitz-Berning cites a newspaper entry in the 'Deutsches Adelsblatt' which had the title 'Fürst Bismarck der erste Nationalsozialist' ("Count Bismarck the first National Socialist").
But the guys we call national socialists have their start in the wake of World War 1. Karl Harrer and Anton Drexler were the initial leaders of what would become the DAP, the German Workers' Party. They had a tiny amount of followers in their native Bavaria, where they were found by a World War 1 veteran who had retained his military post even after his country's defeat in the war - Adolf Hitler. Hitler over-mystifies his entry into the party in "Mein Kampf", where he describes how he shouts down a 'professor' who suggested a merge between Bavaria and Austria in a DAP meeting. Following Hitler's apparently impressive rhetorical display, Drexler approached him to join the party. Whether the story of Hitler reking someone with facts and logic, as the kids these days might call it, is actually accurate or whether Hitler made it up for dramatic effect cannot be decisively said as far as I am aware.
Whatever the case, Hitler joins the DAP after some reservations. He was initially skeptical and assesses that he had planned to found his own party, but ended up changing his mind - he would end up hijacking the DAP anyway.
The party would be renamed by early 1920 and went from Deutsche Arbeiterpartei to carrying a new identifier, making it the Nationalsozialistische Deutsche Arbeiterpartei, the National Socialist German Workers' Party or NSDAP for short. Shortly before, in April, Hitler had cooperated with Drexler and one Gottfried Feder to draw up some 25 points central to what would be renamed to become the NSDAP on 24 February 1920, the same day Hitler proposed the new program.
Union of all Germans on the basis of self-determination of the peoples in a Greater Germany.
Emancipation of the German people in front of all other peoples, revision of the Versailles and St Germain treaties.
Return of German colonies.
Limit of German citizenship to those of German blood. Citizenship is to be denied to Jews.
Those who aren't citizens [see (4)] have to live under legislation for foreigners.
Only citizens are to determine political matters. Only citizens are to hold or run for office at all levels of government.
The state is to serve to feed the people. If not all people can be fed, non-citizens need to be deported in order to secure the food for citizens.
Any immigration of non-Germans shall be illegal. Any non-Germans who immigrated since 2 August 1914 are to be deported.
All citizens are to have equal rights and duties.
The first duty of the citizen is to work mentally or physically. The deeds of the one cannot interfere with the interests of the many and has to serve the greater good.
Removal of income without work. [as a result of (10)] Break the Zinsknechtschaft ['interest serfdom'].
Personal enrichment through war shall be a crime. Any personal gains from World War 1 shall be requisited.
Nationalisation of all trusts.
The state must profit from big corporations.
Old age pensions must be secured and expanded.
A solid middle class must be provided for. Big businesses should be communalized for the profit of small businesspeople.
Redistribution of soil ["Bodenreform"]. Create legislation to take away land from its owners without compensation.
Use the death penalty against those who hurt the greater good. [special focus is given to economic crimes]
Replace the Roman materialistic world order with a German world order.
The state must enable all Germans a higher education. Overhaul all learning plans. Gifted children from lower classes should be educated by state regardless of their economic misfortune.
Protect mothers and children.
Abolish the mercenary force and form a people's army.
Fight fake news. a) only German citizens may publish or work at German newspapers. b) non-German newspapers must be approved by the government. c) foreigerns may not subsidize newspapers. d) newspapers that threaten the greater goods will be banned. Fight hurtful art and literature.
Freedom of all religions that don't threaten the greater good. Endorse positive Christianity without a particular confession.
Form a strong central government.
[After point 25, there is a clarification in later versions regarding point 17 that the NSDAP shall protect all private property that wasn't acquired in illegal ways.]
So that's what the National Socialists understood under national socialism. Some of those points are pretty socialist, almost all of them are rather nationalist.
I advise to read most allusions to big business or foreign influence as coded allusions to the Jews, by the way. However, the word "Jude" only appears once (in point 4) and the word "jüdisch" twice, when it criticizes the "Jewish-materialistic spirit" in point 24 (a juxtaposition to 'positive Christianity'), and once more in the addendum after point 25, when Hitler clarifies that the illegal ways of acquiring land under point 17 is aimed primarily against the Jewish land speculation companies. ["jüdische Grundstücksspekulations-Gesellschaften"].
Already in the addendum you can see that the NSDAP didn't make national socialism as central in the 25 points but became far less overt about it later on. My personal interpretation of this is the passing of the baton from Drexler, who wrote most of the 25 points, to Hitler, who oversaw the party's self-image from 1921/1922 forwards and who quickly introduced the Führerprinzip and made antisemitism the central and only tangible domestic policy point.
As to the differences to Italian fascism you allude to; Italian fascism is also the brainchild of some sort of perverted socialist policy (Mussolini was a socialist well into World War 1) and has nationalistic collectivisms of its own. However, I shall leave the comparisons to the Italian system and the allusions of national socialism as fascistic (or what at the time would have been considered fascistic, the term wasn't known in Germany either way) to the well-read experts on Italian-style fascism.
Hitler, Adolf: "Mein Kampf", 1923
Kershaw, Ian: "Hitler", 2008
Shirer, William L.: "Rise and Fall of the Third Reich", 1960
Schmitz-Berning, Cornelia: "Vokabular des Nationalsozialismus", 1998
Wucher, Albert: "Marksteine der deutschen Zeitgeschichte 1914-1945", 1991
Not to discourage further discussion, but there is a section of the FAQ on this subject: How socialist was National Socialism? with answers from u/kieslowskifan and u/g0dwinslawyer.