In a comment here, Valorum states "Zero elephants. Several Oliphaunts". Which made me wonder: is there anything in Tolkien's writings to support the notion that Oliphaunts are not elephants?
The description we have is as follows:
Grey as a mouse,
Big as a house,
Nose like a snake,
I make the earth shake,
As I tramp through the grass;
Trees crack as I pass.
With horns in my mouth
I walk in the South,
Flapping big ears.
Beyond count of years
I stump round and round,
Never lie on the ground,
Not even to die.
Oliphaunt am I,
Biggest of all,
Huge, old, and tall.
If ever you’d met me
You wouldn’t forget me.
If you never do,
You won’t think I’m true;
But old Oliphaunt am I,
And I never lie.
Lord of the Rings, Book IV, chapter 3 - The Black Gate is Closed
To me, that says "elephant".
There is also a description of the opliphaunts carrying "towers" on their back. That is reminiscent of depictions of Hannibal's war elephants, for example:
(War elephants depicted in Hannibal Barca crossing the Rhône (1878), by Henri Motte. Source: Wikipedia. Image in the public domain)
Is there anything to contradict the supposition that "Oliphaunts" is what the hobbits call elephants, plain and simple?
(Note I am emphatically not asking about the movie. Peter Jackson decided to make a bunch of changes because that fitted his artistic vision, that's his prerogative.)