Before I retired it was always jag (Scotland) or injection. Usually when giving it to someone after preparing them, "Right here's the wee jag".
Jab to me means someone poking you.
When speaking about it on TV I think it should be referred to as a vaccination because that is what it is.
When nursing the jags I had to give included vitamin B12, flu, pneumonia, hepatitis A & B, shingles, methotrexate, heparin, vitamin D, swine flu, testosterone, goseralin etc etc When there is something new it is better to refer to it as what it is not as just a jag or jab. Many people who have the covid vaccinations will also be having other injections be specific.