What you see here are two typical Italian panini. Notice that they don't look much like what Americans mean by that term.
First of all. Please do not say "a panini." The word, like many other food words in Italian ending with -i, is a masculine plural. I get cross when I hear this. Please say panino.
The word panino, strictly speaking, refers only to the roll of bread itself, whereas panino imbottito (stuffed roll) is technically speaking the sandwich. Italians will not use the latter unless there is some possibility of confusion.
Your run-of-the-mill Italian panino is not grilled. I may be cynical- no, I am cynical- but I think that in the US our panini are always grilled because the crusty bread is not that good or fresh. The grilling hides a multitude of sins. Your average Italian panino, which I much prefer, is fresh, has one or two ingredients, and usually no condiments. In fancy bars and Autogrill (those places on the autostrada that serve food and drink and have gifts), you will have more complicated panini.
Here are some standards: thinly sliced mortadella; mortadella and provolone (or other bland semi-soft cheese); ham; prosciutto; ham and artichoke hearts (love it); cheese alone; salami or other cold cuts. If the bread and the filling are very good quality, you don't need anything else.
Yet another reason why the Italians are slimmer.