usa – My COVID-19 vaccination card doesn’t have the date when the vaccine was administered. Is it ok if I write it in myself?

I received the Johnson & Johnson COVID-19 vaccine in the United States. The provider gave me a CDC vaccination card which lists my name, date of birth, vaccine name, lot number, and name of the provider. But they didn’t fill out the date in the third column of the card. Now I am planning to travel internationally and obviously I want to prove that I am fully vaccinated. Is it ok if I write in the date myself? I’m worried that immigration officials will notice that the handwriting for the date differs from the handwriting on the rest of the card. If they are convinced that I have forged a vaccination record, I could face severe consequences. But some of my friends have told me that some providers expected the patient to fill in the date themselves since they were too busy to do it. If that’s the case, then it seems like there’s nothing inherently dishonest about also filling in the date on my own card.