The term "after-born child" is a legal term of art in the world of trusts and estates law. As discussed in the link posted, the after-born child might change the way a will or trust is interpreted. I have no knowledge of the origins or meaning of the latin term but I would guess it relates to the same set of inheritance issues. Interestingly, the term agnate in English now means a relative related through the father's side of the family, similar to the term enate, which means a relative related through the mother's side.

What you are talking about comes, indeed, from the "Roman law" but the Roman law itself did not just appear out of nothing. So they did not just look back for unresolved issues and decided to include a fix.

Creating the law was more about looking at the existing practices, which were numerous and various, and create one law, based on certain philosophy and values.

At first the Roman law was quite closely bonded to religion and very abstract. The first legal text was "The Law of the Twelve Tables" which appeared around 5c BC, after a plebeian tribune named Terentilium Arsa proposed that the law shoud be written, in order to prevent the magistrates from applying it arbitrarily. The patricians agreed, and sent a delegation to Athens, to copy their "Laws of Solon". This we know due to a Roman historian named Livy, who wrote a history of Rome.

The Twelve Tables was not as detailed as todays laws, did not create a coherent system, and did not give solution to all possible cases. They were merely an improvement of the customary law, with emphasis on private law and civil procedure.

The "agnatus" came to be detailed much later, by a jurist named Gaius, around 160AD. Gaius brought a precise and detailed system of private law, where everything was divided between "persons", "things" and "legal actions". His work used and expanded upon the Twelve Tables.

Up to that point there was a sort of a chief official, the praetor, who would give guidance about each case just before it started, using a selection of very rigid formulae. Gaius mused that better justice could be delivered if that preator's personal guidance could be replaced with clear definitions.

Gaius wrote a textbook called "Institutes",treating the "persons" and their different status within the law, and the "things" with the rights which persons can acquire over them. This included the law about wills and inheritance.

There were further divisions within these "persons", and that's where various "agnati" appear. The agnati were a type of "cognati", or relatives according to the civil law, descended from one person. Since there was a difference wether a person was related through birth, or through a legal link, the latter were called "agnati", with Gaius giving us the precise legal definition. When you have more definitions and more terms describing different people, you can also diversify the law and create sub-rules regarding each case. There were no agnati in the Twelve tables and the inheritance went to the "extended family" instead, which was not as precise in terms of who gets what. Sometimes the birth of an additional child could even invalidate the will, which was a much easier solution than deciding what to do with the newly arrived, lacking detailed prescriptions.

And since the unborn child is not yet a "person" when the will was made, they were considered "agnati" and not "cognati". Sometimes laws seem quite different from the "real world" because everything depends upon definitions.