The Put-Outer is how Rowling describes it
He found what he was looking for in his inside pocket. It seemed to be
a silver cigarette lighter. He flicked it open, held it up in the air,
and clicked it. The nearest street lamp went out with a little pop. He
clicked it again -- the next lamp flickered into darkness. Twelve
times he clicked the Put-Outer, until the only lights left on the
whole street were two tiny pinpricks in the distance, which were the
eyes of the cat watching him.
The Deluminator is how Dumbledore describes it.
“‘The Last Will and Testament of Albus Percival Wulfric Brian
Dumbledore’ . . . Yes, here we are. . . . ’To Ronald Bilius Weasley, I
leave my Deluminator, in the hope that he will remember me when he
Note that we don't know what the formal name of the device is!
As Scrimgeour says:
“That is a valuable object,” said Scrimgeour, watching Ron. “It may
even be unique. Certainly it is of Dumbledore’s own design. Why would
he have left you an item so rare?”
What this means is that Dumbledore is probably the only one who has a name for it. It could be that Dumbledore called it the Deluminator, and J.K. Rowling simply described it in practical (and funny) terms as a Put-Outer.
It could also be that Dumbledore called the device a Put-Outer in his mind (remember, he is quirky), but called it a Deluminator in his will. He could have meant it as a joke (making the device sound more impressive than it was), or to affirmatively identify it. After all, he may have many devices that put things out. He may have wanted it to have a distinguished-sounding name if and when the rest of the magical community became aware of it — he was not above pride.
Could Rowling have intended for the device to be called a Put-Outer? Possibly. But this introduces no inconsistency.