You have it pretty much spot on, let me fill in your blanks in reverse order, so as to be in chronological order, and to end with the bit you describe as "awesome".
You were correct not to be surprised; it's Italian and like many English words of Italian origin, relates to music being originally an opera term.
An interesting early use in this regard is from George Grove;s A dictionary of music and musicians:
Scenario, an Italian term, meaning a sketch of the scenes and main points of an opera libretto, drawn up and settled preliminary to filling in the detail.
Interesting, because it shows us that in the 1880s it was being used in English, but still noted as "an Italian term".
Now, you ask:
did it experience another shift of meaning in 1910?
Yes, the cinema! As cinema moved from its early infancy into being a narrative art, it took the term scenario and applied it to what is now called a shooting script.
That sense has now died out because...
So, did "scenario" experience a shift in meaning in 1960?
Yes. As you noted:
it became used to describe nuclear outcomes during the Cold War
More specifically, in the hey-day of the think tank in the aftermath of World War 2, Herman Kahn was working for the RAND Corporation on applying game theory to strategic planning for the potential of conflict between NATO and Warsaw Pact countries with their potential to involve nuclear warfare.
He was heavily involved in producing an approach to thinking about plausible outcomes rather than just those deemed likely, and planning accordingly. His approach involved writing scripts for (so-far) fictional events, as if written by people in the future. Since this was essentially taking the fiction-writer's approach to practical ends, he took the term scenario from the dramatic arts, and so he is considered one of the inventors of scenario planning and almost certainly where it takes its name.
While best known now as one of the inspirations for "Dr. Strangelove" his book Thinking the Unthinkable* had some popular success and was much read by journalists in particular. From this scenario in his specialised use, in which it was essentially a piece of military strategic jargon, moved into the popular culture, and from then became increasingly loose in meaning.
Kahn went on to write crazy predictions of how by the year 2000 we'd have new forms of birth-control, sex-change operations, widespread peaceful use of nuclear power, real-time banking and phones that fitted in our pockets. (Clearly a mad-man).
He'd done his bit for scenario, turning it to a jargon use that mutated into the form we know. Thinking the Unthinkable was published in 1962, which you'll note comes in just before the rise in the graph in google ngram.
*Which amusingly enough cites Red Alert, the novel that Dr. Strangelove was a loose adaptation of. Reading it now it's hard not to picture Dr. Strangelove using Dr. Strangelove references.