What are you referring to by "birth rate"? Are you referring to the general society-wide system of babies being born? Or are you referring to a specific metric relating to how many babies are born?
From definitions I find online, first order vs second order refers to systems. So if you're applying those adjectives to a specific metric, I'm not sure that's correct.
I don't think we can say that people "universally" agree about what metric the term "birth rate" should refer to, because different people in this thread are using the term to refer to different metrics. However, I do think that generally the term "birth rate" refers to the number of babies born per year (or other unit of time), and the term "fertility rate" refers to the number of babies born per woman. Fertility rate is actually somewhat more complicated than that, because it takes into account the different fertility rates for women of different ages, and then sums those together, to simulate how many children a woman would have who passes through all the different ages.
