The quick and dirty definition of fascism:
fas·cism n.1. often Fascism
a. A system of government marked by centralization of authority under a dictator, stringent socioeconomic controls, suppression of the opposition through terror and censorship, and typically a policy of belligerent nationalism and racism.b. A political philosophy or movement based on or advocating such a system of government.
2. Oppressive, dictatorial control.
Source: The American Heritage® Dictionary of the English Language, Fourth Edition
Franco was and is universally recognized as having been a fascist.
Clearly Hitler and Mussolini considered him to be their fascist brother in arms.
The difference between Spanish fascism and that of Germany and Italy was the primacy of the party.
In Germany and Italy the state was subordinate to the party but it was vice versa in Spain.
Not really. In fact Hitler and Franco despised each other, almost viscerally, from the moment they met. Franco thought Hitler was a declasse and irreligious little nihilist who posed a danger to Europe's Christian civilization. He was right.
Hitler thought Franco was a bit too Catholic and stodgy to be a true brother in arms.
But that's beside the point. The dictionary definition of fascism you gave (basically, "fascism=bad") is not all that helpful. It does not, for instance, distinguish fascism from communism.
There was a world of difference between Franco's program in Spain and Hitler's in Germany...so great a difference that I'm not sure the term "fascist" can contain them both.