In the U.S., we get the white Covid-19 card from the CDC. What information is filled out on the card depends on where you get your vaccination.
Some places write down the location, some don't. Some places have a signature of the person administering the vaccine, some dont. In the U.S., we don't have ID numbers.
Our name, and date of birth is entered by hand. Some by the people giving the card, others expect us to do it.
But the one thing our cards do have is a sticker with information about the dose we are given....name of vaccine, batch number and expiration date. The date of vaccine is entered--by hand--next to it.
More than likely, if a country wants American tourists, they will accept this card. They can't expect millions of Americans to return to where they got the shot to get all new cards.
European countries are just beginning to open up. Their guidelines NOW are based on the EU. They need time to figure out what information is available from visitors outside the EU? It's not just the U.S. It's Canada, China, Japan, etc. Are they going to ban tens of millions of tourists because the name of where they got the vaccine isn't available? Or the card wasn't signed by the person giving it?
The easy way is to take whatever record is given and make sure the name and DOB matches the passport. The additional information will show the name of the vaccine, batch number, expiration date and the date the vaccine was given. If it's a CDC card the shot was given in the U.S. Does it really matter if it was given at a CVS in Boise or a drive through clinic at a sports stadium in Des Moines?
The other option is to have people fill out a form to present with the CDC card giving information not on the card.
I know some people here will say these cards can be counterfeited. True. But there is no perfect way to do this. It's too late to go back and tell everyone they have to return to get new info.