How did the word "production" end up being a synonym with the word "rule" in the context of Computer Science?
A grammar has "Production rules:" rules about the new sequences of symbols that you can produce from old sequences. In the cases of context-free grammars, this the old sequence is always a single non-terminal symbol.

Sometimes people abbreviate "production rules" to "productions". Sometimes people abbreviate "production rules" to "rules".

The two are not synonymous: there are plenty of things which are rules but are not productions, and people may call a single use of a rule a production, or the rule itself a production. Generally, the hope is that it's clear from the context what is meant.