Consider the following quote:
'Albus Dumbledore,' said Dumbledore, when Uncle Vernon failed to affect an introduction. 'We have corresponded, of course.' Harry thought this an odd way of reminding Aunt Petunia that he had once sent her an exploding letter, but Aunt Petunia did not challenge the term..
Will and Won't, Half-Blood Prince
I always took this highlighted part of the quote at face value, and assumed that Dumbledore was, in his inimitable style, referring to the Howler and Aunt Petunia was just too shocked and scared to say anything (after all, she was speaking to a wizard, and one who was eccentric enough to dress in complete wizard outfit at that.)
An alternative explanation
However, I've come across a new explanation from another Potterhead that states that Dumbledore was actually referring to
the fact that he and Petunia had corresponded way back, when Petunia was just a kid, and had written to Dumbledore, asking to be allowed to study in Hogwarts, and received his reply denying her admission.
Again, it may be that Petunia was just too flabbergasted to speak, but what differs is the correspondence Dumbledore was referring to.
So, which of the explanations is correct? The first? The second? Or both? Or something else? What the 'correspondence' Dumbledore referred to?
And did Dumbledore correspond with Aunt Petunia anytime else?
Note 1.: I'm referring to Dumbledore corresponding with Petunia alone, not the Dursleys together. He might have corresponded with the Dursleys together a couple of times.
Note 2.: As part of this, I've not counted the letter Dumbledore wrote to the Dursleys explaining why Harry had to be left with them. Nor the letter explaining their severe breach of school rules by flying the Ford Anglia.
Note 3.: Please use only canon sources and back up your answers with evidence.
In searching for communication between Dumbledore and Petunia, I came across this answer
but that doesn't specifically answer my question.