Green Sap, the tower of the Bosmer. Grown, instead of built, I've heard it was Falinesti, the Graht-Oaks, and one UESP, it's the Elden Tree and that the Orrey of Anumaril was a plot to remake Green-Sap into White-Gold and restore the Ayleids. Could someone explain this? Which is correct? I've always understood that Falinesti was Green-Sap.

Michael Kirkbride was once asked what the Stone of Falinesti was. He said it was a fruit. He didn't explicitly say Falinesti was a Tower, but when presented with the premise that it was one, he rolled with it. Hence, many in the community came to the conclusion that Falinesti is Green-Sap.

ESO's Aurbic Enigma 4: The Elden Tree tells us that Green-Sap is the Elden Tree. Theoretically, due to the nature of Green-Sap, there could be some convoluted story similar to that book explaining how Falinesti is also Green-Sap. And hypothetically, Falinesti could be an entirely different Tower altogether. But I'm more inclined to think that the apparent contradiction just exemplifies the unreliability of unofficial sources.

Is the Elden Tree supposed to be the Elden Grove that was in Arena?

The Elden Tree is in the Elden Grove.