What you mentioned in the first paragraph is not vaccination, it is called variolation. It predates vaccination by a large margin and was far more dangerous.
(Variolation used real smallpox, arm-to-arm used the vaccine virus instead) https://en.wikipedia.org/wiki/Smallpox_vaccine#Types
“As medical researchers subjected viruses to serial passage, inadequate recordkeeping resulted in the creation of laboratory strains with unclear origins.[95]: 4 By the late 19th century, it was unknown whether the vaccine originated from cowpox, horsepox, or an attenuated strain of smallpox.”
What conspiracy? It’s a theory of lost knowledge / confusion, but no one is conspiring with anyone.
https://en.m.wikipedia.org/wiki/Vaccinia
There’s also “kinepox”, but I _think_ that’s just a different method of cowpox vaccine delivery.