That is correct. In pan-India exams like the JEE and NEET, and even in high-stakes state-level exams, it would be impossible to cheat so blatantly.
That doesn't mean cheating doesn't happen, but it would require you to know someone in the examining body. And, they would be risking their careers and their liberty to help you.
And, I know a few people who were vaccinated, but the dose did not get registered. In some cases, this was purposefully done - so that they could get the dose while they were below the age floor (at that time). In other cases, this was accidental - the hospital staff simply neglected to put it into the system, and no one cared.