As cbornsh2 perfectly explained in their answer, Dumbledore did use the stone incorrectly. That answers the question you highlighted at the end ("What is the correct way to use the Resurrection Stone?"). But if your main question is the title: "Why did he put the ring on?" then maybe there is another answer.
We assume that Dumbledore wanted to use the stone, but what if that is not the case?
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In your quote, Dumbledore states that "through [his] youth [he] had wanted it for very different reasons":
"When I discovered it, after all those years, buried in the abandoned
home of the Gaunts - the Hallow I had craved most of all, though in my
youth I had wanted it for very different reasons - I lost my head,
Harry. I quite forgot that it was now a Horcrux, that the ring was
sure to carry a curse. I picked it up, and I put it on..."
-- Harry Potter and the Deathly Hallows - Chapter 35: King's Cross
His first reason was his and Grindelwald's obsession with the Deathly Hallows, which is of course related to his second, later reason: seeing his mother and sister again after their deaths.
Maybe the reason driving him at that moment is not the one we believe.
The Resurrection Stone was the last Hallow that Dumbledore had yet to find.
We can even argue it was the only one he didn't control : the Wand was his and even though Harry owned the Cloak, Dumbledore had had it up until Harry's first year, and he could easily have gotten a hold of it again (asking Harry to lend it to him or to the Order for example).
"I was fit only to possess the meanest of them, the least
extraordinary. I was fit to own the Elder Wand, and not boast of it,
and not to kill with it. I was permitted to tame and use it, because I
took it, not for gain, but to save others from it."
"You have guessed, I know, why the Cloak was in my possession on
the night your parents died. James had showed it to me just a few days
previously."
-- both quotes from Harry Potter and the Deathly Hallows - Chapter 35: King's Cross
Then, maybe the reason he put the ring on was something else.
When he says that he lost his head, he doesn't mean
"I lost my head, I'm old, I didn't remember how it works."
Remember that he and Grindelwald spent months researching everything there was to be known about these Hallows. Of course Dumbledore, one of the most powerful wizards alive, didn't just forget such a basic piece of information about his old obsession just because of emotion (even Harry remembered it !).
Dumbledore is often described as old but never as senile, forgetful or easily distracted by emotions (that would be quite impractical in duels).
I believe he means
"I lost my head, my old demons came back and for a moment I was attracted to the power of the Hallows again."
We have seen with the Cloak that even after he renounced his earlier beliefs and stopped his quest for the Hallows, they still had power over him.
"I could not resist, could not help taking a closer look ... [...]
and then your father died, and I had two Hallows at last, all to
myself!'
-- Harry Potter and the Deathly Hallows - Chapter 35: King's Cross
Even though we could logically think that he wanted to use the Stone to see his family, I believe that he lost control again, like when faced with the Cloak. And that is why he put the ring on instead of turning it in his hand which he knew was the correct way to use it.
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He put the ring on because what he wanted was not to use the Stone, but to own and control it, as one of the object that could make him the Master of Death.
“I was such a fool, Harry. After all those years I had learned
nothing. I was unworthy to unite the Deathly Hallows, I had proved
it time and again, and here was final proof.”
-- Harry Potter and the Deathly Hallows - Chapter 35: King's Cross