I agree in general with Edwin Ashworth's answer, but perhaps the reader is looking for a less learned explanation. If so, I offer this:
The difference between "One day, after I had my breakfast, the postman came" and "One day after I had my breakfast, the postman came" is that in the first instance "one day" is there to indicate the time frame of the entire action of the sentence: On one and the same day, (1) "I had my breakfast," and then (2) "the postman came." But in the second instance, "one day" is the interval between the two main actions in the sentence: "I had my breakfast," and then, one day afterward, "the postman came."
The difference between the two Rome sentences is far less significant, in my opinion. I suppose that "In the olden days, in Rome, people were nice" separates "in Rome" from direct attachment to the preceding phrase "In the olden days," so that it is arguably free to attach to the following "people" instead. But if there is a substantive difference between the ideas expressed by
In the olden days in Rome, people were nice.
In the olden days, people in Rome were nice.
it is pitched at a level of nuance beyond my poor powers to apprehend. The conclusion I would draw from your two pairs of examples is that sometimes the phrase-splitting comma has a large and fundamental effect on sentence meaning, while at other times its effect on sentence meaning is effectively zero.