You are not immune from the virus, but you are protected from it...
But it depends on the vaccine you get! If it's Sinovac, it seems that you are only 50% protected, if it's AstraZeneca it's 80%, if it's Moderna or Pfizer it's more than 90%! But keep in mind that most of these vaccines will give you protection only after the 2nd injection.
In Israel, where almost all the population got vaccinated twice, the cases were reduced by 92%, meaning that it works!
We don't know yet, for sure, about the different variants of the virus. Up to now, only Pfizer and Moderna show to be effective with variants like the British one.
And scientists don't know yet for how long time this protection will last! Maybe one year or maybe more... we still have many things to learn!
It seems that very young children like yours, have their own antibodies that can defeat the virus which is good news!
In any case, you need to always wear a surgical mask (they are better than cloth masks). Cloth masks have been proven to be useless against the Covid-19 variants.
Take care everybody!