Oh my goodness. It is nothing like gene therapy and your colleague is talking rubbish
I think we need to be careful here because there is definitely "scientific debate" surrounding this.
Saying things like this just... well, what good does it do? Have you ever changed someone's mind by telling them explicitly (and without any further explanation) they're talking rubbish?
Understandably, the people who make these important decisions have decided that calling it gene therapy applied to vaccines would put off too many people, due to the negative perceptions of gene therapy.
So they've distanced themselves from that.
But the people who believe this can pretty easily find the inventor of mRNA vaccine technology saying that both vaccines use gene therapy technology. He goes on to say that the people in public health messaging don't like it - because it turns people off - but "I'm a scientist".
It's also easy for these people to find out that the FDA themselves considered mRNA gene therapy.
And if you look hard enough... you'll find pharma companies like Moderna arguing in their initial offering under risk factors that: Unlike certain gene therapies that irreversibly alter cell DNA and could act as a source of side effects, mRNA based medicines are designed to not irreversibly change cell DNA; however, side effects observed in gene therapy could negatively impact the perception of mRNA medicines despite the differences in mechanism
So someone could connect the dots (because remember humans always want to connect the dots) and say that both the scientists and the FDA considered mRNA to be gene therapy, but the pharma companies want the definition of gene therapy to apply specifically to only the certain gene therapies that irreversibly alter cell DNA, due to negative impacts on public perception.
The problem is that rather than just being honest and actually educating people, you have the "fact checkers" and the "what a load of rubbish!!" Neither of which do anything at all to actually explain what has happened or why it's happened.
So it gets turned into this "smoking gun" where people think they're hiding things, they're in the pockets of big Pharma, they're suppressing information etc etc until it grows legs and runs away.
Why not just be honest with people? 
Give those who are smart enough to do all that research above (which I found within about 2 minutes on reddit) the actual information... and give the ones who don't really care the simplest of definitions / analogies like the one I used below.
"The internet was first used for the military"
"mRNA technology was first considered gene therapy"
"The definition of communication changed with the development of the internet"
"The definition of vaccine changed with the development of mRNA technology"
Smoking gun averted, conspiracy theory explained with logic, intelligent thought (which includes questioning, imo) can resume?