MIC card

I believe that the MIC card is only available for permanent residents of Rome, so unless you are living there, you would not be eligible. https://www.museiincomuneroma.it/en/infopage/mic-card

From the website:
You can use the MIC if you are:
- an adult resident in Rome or in the Città Metropolitana of Rome
- a student at public or private universities in the area of the Città Metropolitana of Rome
- an adult domiciled in Rome or in the Città Metropolitana of Rome

Because the first option is an adult resident, I assumed that the third option was for someone staying in Rome. When I reached out the the help email address, they sent me to an all italian video, so I still don't know. My email indicated that I was from the US and they did not correct me. This is why I still had the question.

M&M, to domicile means to live there; not visit. You may not necessarily be a permanent resident but you would not be considered a visitor either. For example, someone may have a home in Florence and that is their legal residence, but they spend 6 months out of the year in Rome. Rome would be considered a domicile. It's similar to a Florida snowbird, who spends winters in one place, but lives in another.

domicile

: a person's fixed, permanent, and principal home for legal purposes

resident

to dwell permanently or continuously : occupy a place as one's legal domicile

"resident' and 'domicile' are in bold on the MIC website, indicating/implying they are defined terms

Above definitions from merriam-webster.com

My interpretation is that a person subject to the 90 day Schengen rule is not a 'resident' nor would be considered to be 'domiciled' in Rome

I have no direct answer to your question of 'which address' to enter, but I can foresee a situation where when you show up at the door and display your 'tourist aura' could be asked for your ID ( "proof of residence") , your US passport will likely betray you are not a 'resident' of Rome.

The MIC card is for residents of Rome including students, but not tourists. The low cost of 5 euro per year is to encourage locals to visit their museums. I needed to show proof of residency when I applied for mine.

Thank you. That all makes perfect sense.