I really need to stop this habit of pinned-topic-neglect...
The term "Elder System" exist in neither the Japanese nor the English version.
In the English version, the bits referencing the Elders are spoken and do not exist in text. Whether Sera refers to their system as "Elder System", "Elder's System", or "Elder's system" cannot be clearly distinguished.
The fact that Yuna doesn't refer to it in such strict terms means very little, given that the Master System is also informally referred to as simply "the System" by her, Sera, Gatz, and the Master.
Incidentally, this same uncertainty exists with the systems of Elysium and Terra when Yuna speaks of them after the battle with Geetz.