While this doesn't answer the meat of your question, I have posted on this sub before about the names of the world wars. In American English certainly, the terms "the World War" and "the Great War" were interchangeable from the time the first war broke out in 1914, and "World War" had essentially become the more widely accepted term by the time it ended in 1918. In the early 1920s, well before the events of the second war had begun to shape up, "World War" was already far and away the preferred term in the United States, while "Great War" had largely fallen out of favor.

In British English, "Great War" stayed more popular for much longer, but "World War" was a well-understood alternate even during the war. The latter term ultimately overtook "Great War" there, too, before WWII began, though not until Hitler's rise to power, or thereabouts, a decade or so after the U.S. had made the change.

In another post, I gave some details about how predictions of a "Second World War" were being made even before the first one was over, and these predictions were made regularly throughout the interwar period, due to what many considered to be an unsettled peace. Once the second war did begin, the press in the English-speaking world readily adopted "Second World War"/"World War II" as the name, from its earliest days.

But any more details as to why these predictions were being made, or why the peace of WWI was considered tenuous is beyond my scope of expertise. Maybe someone else can chime in with a more detailed response on that aspect of your question. Nevertheless, whether the English language had settled on "Great War" or "World War" for the first conflict, I suspect the rhetoric about a second one coming would have remained the same during the interwar period. We'd just be talking about the "Second Great War" instead.

And, notably, other languages had their own terminology. As far as I know (and being a native German, you may know better than I do), the first World War was always called the "Weltkrieg" in the German language, and nothing else ever really rivaled it as an "official" name. Designating it as the "Erste Weltkrieg" came at pretty much the same time as in the English-speaking world. That is, once Germany invaded Poland in September 1939, and the UK and France responded by declaring war on Germany, the German press quickly acknowledged the events as shaping up to be a "Zweiter Weltkrieg", and so the original "Weltkrieg" was retroactively designated as the "erste" one.

That isn't to say there weren't detractors from this terminology. As detailed from a U.S. perspective in my second post linked above, there were isolationists in the U.S. who denounced the term "Second World War" as inviting a global response at a time when it was confined to Europe (nevermind that Asia had war already going on at that point).

The Roosevelt administration was leery of using the "Second World War" terminology early in the war, because American involvement in the first one had been unpopular, and they didn't want to scare off a large portion of the American public. They initially were promoting patriotic names like the "Survival War", the "War For World Freedom", and the "War Against Oppression". Respondents to a 1942 Gallup Poll additionally suggested "War for Liberty", "Anti-Dictator War", and "War for Humanity" as alternates to "Second World War", which was firmly rooted as the name of the war by then. Detractors suggested terms like the "Rich Man's War", the "Meddler's War", and the "Roosevelt War".

But again, the meat of your question seems to be about how the public perception was/has been affected by naming these wars in this way. That's a question I don't have an answer to, so maybe someone else can add something here.