In my mind, these two sentences are not necessarily synonymous. “John has an ability to make money” says to me that he is or has been successful, in general, in making money, doing whatever. “John has the ability to make money” tells me he is capable of making money but who knows if he actually has made money.

First, in this case as in many others, you can use either “the” or “a” without it sounding too bad. Which of the two is (marginally) better has to do with how definite you want to sound. Superman has the ability to fly. (It’s clear and well-known) Jessica Jones has an ability to jump high. (Not as clear, more ambiguous) As far as “ability to make money”. “Ability “ isn’t the usual word to use. By default, making money is something you do in return for doing something else. It’s not intrinsic to the person and not noteworthy. So what are you trying to convey by describing it as an ability? These are much more significant than the debate over the/a. John can earn money. John can make money. John has the capability to support his family, if only he’d stop drinking.

I suspect you are overthinking this. It means merely that John can make money, but it does not say what that skill is specifically.