If I overstay in the Schengen area, which country would impose a fine on me?

Difficult to definitely predict what will happen but Germans have a reputation for being particularly strict and my personal experience bears this out so if this is at all possible try to avoid falling fool of the rule.

As @Gagravarr explained, there should be no systematic passport check when landing in Germany on the return flight from Italy but your passport will be checked when transiting to the international zone at the airport in Munich/boarding a flight outside of the Schengen area and that’s where it will be stamped/checked for potential overstay. The German border police will consider your stay in the whole Schengen area and the date you first arrived in any Schengen country, so they can also impose a fine even if you stayed in Italy for most of the time. The Schengen area can indeed be considered as one country in this respect, the fact that you stayed less than a day in Germany or regard this part of the travel as transit does not change anything. The departure date from the US does not count either.

I don’t understand the advice about a “change of date fee”. As far as I know, there is no easy way to extend a Schengen stay for a fee, the three-months-in-any-six-month-period is a hard limit (a fine could always be imposed of course but that’s not an extension fee, you are still staying illegally and supposed to leave immediately). What you would need is a long-stay visa, which is an entirely different kettle of fish and something many EU countries don’t even deliver at all to people who already entered the country.

In terms of practical steps to stay out of trouble, there are three things you could do (from most to least practical):

• As @Gagravarr suggested, leave the Schengen area for a few days between now and your departure date, e.g. go for a week-end to Croatia (from Trieste, this is really close) or to the UK (many budget airlines fly from the UK to the South of Europe and tickets from Italy/Spain to the UK at the beginning of a week-end and back might even be cheaper than the other way around). That way, your stay in the Schengen area will be under three months.
• Change the date of your return flight (probably expensive and inconvenient but it would also obviously solve the problem!)
• Travel to Italy by train (it would not make the whole thing legal, just reduce – not eliminate – the risk of having your passport checked before boarding the plane to the US). Of course, you would still risk a fine in Munich and your return ticket would also need to be changed because failing to use the first leg of the trip could cancel the next one so this is unlikely to be easier or cheaper than just flying on another date and doesn’t even solve the overstay problem. But at least you would not need to worry about being denied boarding in Italy.

EDIT: I understand now that you are not in Europe yet and you are also concerned about being able to board the plane on your onward journey. The reason the airline might refuse boarding is that they have to fly you back to the departure point should you be denied entry at the destination and they don’t want to take any chances. Also, once you are at the airport ready to depart, you depend on them so even if some random Italian police agent told you on the phone that you will be OK, you still need to convince the handling personnel that this will not be an issue. Even if they were demonstrably wrong and you could somehow obtain some form of compensation later on, if that’s not the way they understand the rule, you will not be able to board the plane, which would obviously be a major inconvenience. The safest course of action is therefore to simply heed their advice.

You seem to be desperate to find someone who will tell you that there is some ways to bend the rule to conclude that you are staying less than three months but as far as I can tell, it’s not the case. At the end of the day, one or two extra days might be OK (reports on various forums suggest that Italians might be a bit more accommodating than Germans in this respect) but you can’t expect bureaucrats to encourage you to break a rule or officially tell you it doesn’t matter.

Also, people everywhere are having a very hard time crossing borders for all sorts of reasons and, often, for no particular reason at all. It’s the way the world works today. Objectively, if you can come to Europe for three months with no visa or paperwork, you are already extremely lucky. Most visitors to the US also have a harder time coming in. It’s frustrating but the only way to make sure you avoid any problem at this point seems to be to change your ticket.

Finally, a small note: Unless you change your plans, just forget about the Italians. In your scenario, you need to deal with Lufthansa and the German federal police. You will enter and leave the Schengen area in Germany and that’s the people who will look at your passport/visa situation.