MTG: is exile the equivalent to leaving the battlefield?

Nope, it means what it says: Leaves the Battlefield.

If it dies, it leaves. Exiled? Leaves. Bounced to hand? Leaves. Shuffled into the library? Leaves.

Here's the specific rules:

603.6c Leaves-the-battlefield abilities trigger when a permanent moves from the battlefield to another zone, or when a phased-in permanent leaves the game because its owner leaves the game. These are written as, but aren't limited to, "When [this object] leaves the battlefield, . . ." or "Whenever [something] is put into a graveyard from the battlefield, . . . ." An ability that attempts to do something to the card that left the battlefield checks for it only in the first zone that it went to. An ability that triggers when a card is put into a certain zone "from anywhere" is never treated as a leaves-the-battlefield ability, even if an object is put into that zone from the battlefield.

Leaves the Battlefield

A permanent "leaves the battlefield" when it's moved from the battlefield to another zone, or (if it's phased in) when it leaves the game because its owner leaves the game. See rules 603.6c and 603.6d.

(the other zones are:

400.1. A zone is a place where objects can be during a game. There are normally seven zones: library, hand, battlefield, graveyard, stack, exile, and command. Some older cards also use the ante zone. Each player has his or her own library, hand, and graveyard. The other zones are shared by all players.)

Not equivalent.

"Leaves the battlefield" means just that. Any time the card moves from the Battlefield to any other zone of the game, be it the Graveyard, the Hand, the Library, the Exile Zone, etc.

If it meant "goes to the graveyard", then it would use the wording "Dies". Leaves the battlefield means it could go to exile, your hand, the graveyard, or to your library.

yes, leaves the battlefield means it goes from the battlefield to another zone (can be to the graveyard, exile, or hand.)