There's so much mis/dis-information around this stuff I think it's worth being precise. Otherwise you get people arguing with each other where neither side is correct and it's strawmen all round.

I get it. We've all been taught vaccines are all variations on a theme, but I think most have taken the wrong bit. As you say, the theme is training the immune system on something. It's not where that something comes from.

There's people who are scared of vaccines because "they're rewriting our DNA". It's crap, but it's because someone heard about mRNA. They've misunderstood and become alarmed. That then starts spreading. Telling people "You've misunderstood this new amazing tech" lands a lot better than "That's not even how vaccines work" and then explaining something that isn't the COVID vaccine.