No, fertility doesn’t care, it is a state. When one conflates birthrate with fertility they end up arguing semantics...
As one of the definitions of fertility we do consider birthrate, but I would say that is a bastardization of the true intent of the word (a state or the ability to be fertilized / produce offspring).
Why consider birthrate as a sub-definition of fertility when birthrate perfectly well explains itself as does fertility? Like I said, it is a conflated term especially considering all of the other definitions speak directly to the ability to produce offspring or support food production as in soil.
fer·til·i·ty
[fer-til-i-tee] Show IPA
noun
1.
the state or quality of being fertile.
2.
Biology . the ability to produce offspring; power of reproduction: the amazing fertility of rabbits.
3.
the birthrate of a population.
4.
(of soil) the capacity to supply nutrients in proper amounts for plant growth when other factors are favorable.
So you’re quibbling with meaning 1 over meaning 3.
Both are legitimate uses of the word by your own source.