Vaccine the word came from vacca for "cow", and cowpox virus being weakened and injected to prevent cowpox.

MRNA is not weakened virus, it's a spike protein in this case.

Or so that's the argument.

Cowpox was used to vaccinate against smallpox. Cowpox is a less severe infection that stimulates the same immune response as smallpox, but it’s not the same virus.

Inoculation is where a weakened pathogen is introduced to create immune resistance against that same pathogen (see inoculation parties).