as long as vaccines are optional it should be fine. But calling them vaccines is an overstatement
So yes, covid vaccines do that.
Yes, the CDC changed its definition of vaccine to be 'more transparent' https://www.king5.com/article/news/verify/coronavirus-verify...
> [T]he CDC changed its definition of vaccine from "a product that stimulates a person's immune system to produce immunity to a specific disease, protecting the person from that disease" to "a preparation that is used to stimulate the body's immune response against diseases."
The mechanism of a flu or covid shot is exactly the same as any other vaccine; the only difference is the degree of efficacy due to human mucosal immunity not having evolved to fully block infection from respiratory viruses.
https://journals.plos.org/plospathogens/article?id=10.1371/j...