Yes all tests of the allowed vaccines need to proove, that taking the vaccine has far less harmful results then going through the infection it is supposed to pretect.
And I doubt, that there is a vaccine available legally, that does not have proof for this to be true.
What is a vaccine actually? Is there a medical definition?
"A vaccine is a biological preparation that provides active acquired immunity to a particular infectious or malignant disease."
https://en.m.wikipedia.org/wiki/Vaccine
Clearly there is addition information in the article, if you are willing to read. But I think this phrase serves as a generalized definition.
Second source:
"a preparation that is administered (as by injection) to stimulate the body's immune response against a specific infectious agent or disease: such as"
