Sorry, I think you're right. A vaccine, legally, only has to prevent disease. That doesn't mean it has to cause your body to sterilize/kill the virus. That, and immunity is not an on/off thing but a scale in how much of a disease your body can prevent.
But then I'm confused because couldn't we then consider, for example, vitamin D a vaccine? It's more commonly understood as a prophylactic - but then what's the difference between a vaccine and a prophylactic?
Just that vaccines are traditionally meant to be made from dead viral agents?
EDIT: Okay I found the difference:
Here's the definition of vaccine from 2017: https://web.archive.org/web/20170221053411/https://dictionar...
And here's the definition right now: https://dictionary.cambridge.org/dictionary/english/vaccine
So, the definition changed from "[an innoculated bacterium/viral agent] that [prevents disease]", to "[any substance] that [produces antibodies]".
A prophylactic is "[any substance] that [prevents disease]".
Technically that still means a prophylactic is not a vaccine and vice versa because of the requirement that vaccines generate antibodies specifically.