Probably more correct to say your wife is 'vaccinated,' and your baby is 'immunized' (or potentially immunized).
Here's from the CDC:
"Vaccination: The act of introducing a vaccine into the body to produce immunity to a specific disease.
"Immunization: A process by which a person becomes protected against a disease through vaccination."
"Recent reports have shown that people who have received COVID-19 mRNA vaccines during pregnancy (mostly during their third trimester) have passed antibodies to their fetuses, which could help protect them after birth."
The vaccine itself doesn't reach the baby, which seems to be the definition of the baby having been "vaccinated." What reaches the baby are antibodies the mother produced herself, in reaction to having gotten the vaccination. Those antibodies (which are not present in the vaccine but were created by her) reach the baby and provide the protection.
(Actually, this is pretty similar to when I got a flu shot when I was pregnant. The contents of the shot don't reach the baby, but if I had gotten the shot in December and the baby was born in January, presumably the baby would have some antibodies to that year's flu in his bloodstream.)
I'm not sure you have the vaccinations right. There are several different ones, and we don't know which one the mother got. Two of them work on Messenger RNA, not DNA. It's your immune system that is being told what to do. Other vaccines are using other methods of getting to the same point, and right now there are a bunch of vaccines out there. The fact the baby developed antibodies doesn't mean they are as strong at preventing covid as getting a vaccine, as the word "could" indicated. I'm guessing we don't know yet. Even the best of the vaccines statistically aren't 100% effective, and there are people getting covid who have been vaccinated, so I'm also guessing the chances of the baby getting covid if the child grows up, never gets vaccinated, and covid sticks around, will be higher than if the baby was directly vaccinated. We also don't know how long antibodies last or even if antibodies are the most important thing the vaccines are doing, as training T cells or just training the body's memory might be stronger and last much longer. Doesn't matter what you call it, really, what matters is how much protection does the baby have upon being born? And that will probably mean a vaccination. Peace.