It was a period of armed conflict following an agreed peace treaty, so a separate conflict from the Second Punic War.

I think the Roman advocates of the Third Punic War would agree with you assessment that it was finishing a job that had been started in the Second Punic War. That being said, there was an official treaty in between the wars which was upheld by both parties for fifty years, there was a separate declaration of war for the Third Punic War, and both sides were significantly changed from the ones that existed at the end of the Second Punic War (Rome had vassalized Greece, Carthage had lost most of its remaining holdings to the Numidians). In fact the gap between the First and Second Punic wars was shorter than between the Second and the Third. I think that's enough to consider them separate wars.

With conflicts like the Hundred Year's War, there wasn't a formal peace signed in between the phases of the war, even though there were whole decades without fighting. I think that's the big difference why the Punic wars are considered as a series of distinct wars rather than an extended 100+ year conflict.

"Only" fifty years? That's rather longer than, for example, the span separating the Great War from the Second World War. Much longer than between the first and second Gulf wars. Longer than between the Franco-German war and the Great War. Most of a man's lifetime; nobody making high-level decisions in the Second Punic War would be alive by the Third - even the ones who were young soldiers would be getting up in years.

I think you are falling prey to the telescoping effect of long timespans: The span between Alexander and Caesar somehow doesn't feel as long as that between Washington and Obama, although in fact it is of course much longer. Nonetheless, fifty years were still fifty years even before Christ was born.