In the specific case of Italian neorealism, it might be best to think of "neo-" as denotative of something like "contemporary", in that there is no direct "revival" of an antecedent being referred to. In literature, you might look to Italian verismo as a precursor, and in fact "neorealism" was first coined in Italy as a literary term in the 1930s term to describe writing that rejected lyricism and introversion for the existential, dramatic material of everyday life. But that is a different story, and even if there were similar motivations between the development of Italian neorealism in literature and film, there wasn't really a close interaction between the two.

So with all that said, the term "neorealism" is just speaking to the simple fact that there are general aesthetic qualities of realism that change over time. We might not think of a movie from the 1930s as being realistic by our modern standards, but it might have been the height of realism at the time. Italian neorealism both refers to a group of filmmakers and the period (postwar Italy of the 40s and 50s) in which they worked. It doesn't really refer to a "trend"' in filmmaking, it was a deliberate aesthetic commitment taken up by a circle of filmmaker-critics who defined what they were going to do as neorealism (which answers the second part of your question, yes, the label arrived at the same time as the films).

The conditions of the film industry after the war changed the way these directors approached the aesthetics of realism in filmmaking. For one, there was a lack of infrastructure for film production after Mussolini's fascist regime (which neorealism was formed from and in opposition to) fell from power. Many of the film studios had been damaged or destroyed. The visibility of urban ruination after wartime bombings meant there was no way to escape depictions of the hardship and poverty in Italy before its gradual economic recovery.

As a result, directors took their cameras into in the streets, and often worked with non-professional actors. They also often used long takes evocative of the ideal of a Bazinian "pure cinema". If the montage theory of Russian formalism, so often demonstrated in Hollywood filmmaking, manipulated reality to construct the image of a new one, then this was the other way around. The long take, while still consciously selected out of the continuity of reality, is meant to stand for itself, not to be part of a montage which gives it meaning only in juxtaposition with other elements that if removed would render the shot unable to stand on its own.

That's the gist of it! Let me know if you have any further questions and/or would like sources for further reading.