According to my Webster’s Brief International Dictionary (1895):
Vac’ci-nate …( v.t. [verb. transitive.]) To inoculate with cowpox, by virus from cows [the French for “cow” is “vache” – hence “to vaccinate”].
Originally and at the time (1895), the word “vaccine” did not have its own entry in the dictionary, and was given instead under the definition of “[to] vaccinate” as being (solely) the adjective-form of the verb “[to] vaccinate”:
“Vaccine…a. [adjective] Pertaining to cows, or to Vaccination.”
So at the time, cows were or could be infected with cowpox-virus, while humans were injected with vaccine cowpox-virus. The actual substance that was injected into a human is cowpox-virus. If it was obtained directly from a cow, then it is a vaccine cowpox-virus. Literally-translated vaccine means cow-sourced.
Today, the masses have been habituated into using the word “vaccine” as if it were a noun, instead of properly as an adjective to describe the source of (what was originally) the injected cowpox-virus.
By close analogy, if someone you know has, say, a red truck and a blue truck, and you were to describe them to another as, for example, “He’s out driving his red.”, then both of you may be able to reason-out what you mean, but it still technically gibberish.
That is how we have taken the adjective “vaccine” used as a modifier of “cowpox-virus” to indicate that it was obtained directly from an infected cow (and not otherwise) and then injected into a human, and step-by-step converted it into a noun by habituated-use and implied (but false-in-fact) mutual-agreement, to cover any and all of the known and unknown surprise-package-stuff in “Uncle-Tony’s-Joy-Juice” selection.
I have no idea whether the modern vaccines have anything actually vaccine in them (i.e., whether there is any cow-sourced cowpox-virus in them).
By 2030 we may be back to the source-format of 1895 and asking each other: “Have you had your shot of Bio-Weapons-Lab-42 yet?