From another question in this Italy forum, it seems that the paper CDC card works fine for verification of being vaccinated. The name on your airline reservation is supposed to completely match the name on your passport. If the CDC card has either omitted the middle name completely (not even an initial for myself) or only listed a middle initial (my wife) while our passports each have our full middle names listed - could this be a problem? I'm thinking in particular for initial entry into Italy.
Perhaps I am being a bit paranoid here, but trying to anticipate any problems. When my wife and I were vaccinated in March 2021, it never dawned on me to insist with our health care provider that our full names (matching our passports) be recorded on the CDC cards. I would probably have to hassle with our health care provider to get our full names on the cards at this point.
We are scheduled for an RSE Heart of Italy in 9 Days tour in late May 2022.
Thanks for any info or experiences.