What "actually" happened is that the First Foundation concluded that the Second Foundation was located on Terminus. One of the main bits of evidence leading to this conclusion is an enigmatic comment by Hari Seldon that he founded another Foundation (other than "Encyclopedia Galactica Publishing Foundation #1") at "Star's End". No one knows what "Star's End" is (in the book there's a visit to a planet named "Tazenda" on the theory it's a corrupted form of "Star's End"), but Arkady Darrell realizes that the Galaxy's rim (where Terminus is located) is a circle and "circles have no end." So "Star's End" is right at the start.
This deductive leap is proven correct when a nest of Second Foundationers is discovered on Terminus.
Except that it isn't correct. Hari Seldon was speaking as a sociologist, not a geometer and "Star's End" is the opposite social end of the Galaxy from Terminus: Trantor. And Preem Palver, a farmer from Trantor, is revealed as the First Speaker of the Second Foundation having sacrificed fifty members of the Second Foundation to fool the First that it had found and destroyed the Second.
"Star's End" as the location of the Second Foundation is first mentioned at the end of the very first Foundation story which later appeared as Part 1 "The Phychohistorians" in //Foundation//, the fix-up novel.
Seldon said, "...a scientific refuge will be established on Terminus. And another will be established at the other end of the Galaxy, let us say," and he smiled, "at Star's End."
Later at the end of Foundation and Empire, Part 1 there's a conversation:
"Then," said Forell, "there are no more enemies."
Barr was thoughtful. "There's a Second Foundation."
"At the other end of the Galaxy? Not for centuries."
Then in "Part 2: The Mule" a character says:
"I know that. Toran and Bayta are leaving with you for that very reason. And, Ebling, there's another, greater purpose. Hari Seldon founded two Foundations three centuries ago; one at each end of the Galaxy. You must find that Second Foundation."
There's a lot more discussion about the Second Foundation's location, but it all seems to be elaborations of this basic information.