>> In 1736 I lost one of my sons, a fine boy of four years old, by the small-pox, taken in the common way.
>> I long regretted bitterly, and still regret that I had not given it to him by inoculation.
>> This I mention for the sake of parents who omit that operation, on the supposition that they should never forgive themselves if a child died under it; my example showing that the regret may be the same either way, and that, therefore, the safer should be chosen.
> Until the very early 1800s, inoculation referred only to the practice of variolation, the predecessor to the smallpox vaccine. Edward Jenner introduced the latter in 1798, when it was called cowpox inoculation, or vaccine inoculation (from Latin vacca = cow). Smallpox inoculation continued to be referred to as variolation (from variola = smallpox), whereas cowpox inoculation was referred to as vaccination (from Jenner's use of variolae vaccinae = smallpox of the cow).
The article goes on to explain how the meanings of the terms have since expanded: https://en.m.wikipedia.org/wiki/Inoculation
There is evidence for such cross-immunity, but not strong enough for this to be a strategy you'd bet on.