Riddle probably wrote in it
The original quote from Chamber of Secrets might seem to imply that Voldemort made the diary into a Horcrux for the express purpose of opening the Chamber.
"Well, he certainly kept an annoyingly close watch on me after Hagrid
was expelled,” said Riddle carelessly. “I knew it wouldn’t be safe to
open the Chamber again while I was still at school. But I wasn’t going
to waste those long years I’d spent searching for it. I decided to
leave behind a diary, preserving my sixteen-year-old self in its
pages, so that one day, with luck, I would be able to lead another in
my footsteps, and finish Salazar Slytherin’s noble work.”
—Harry Potter and the Chamber of Secrets
This may be. However, Voldemort may have purchased the diary before opening the Chamber in his fifth year:
Harry saw at once that it was a diary, and the faded year on the cover
told him it was fifty years old. He opened it eagerly.
—Harry Potter and the Chamber of Secrets
This does not mean that it was literally fifty years old, merely approximately so. In any case, even if he purchased it the year he opened the Chamber, he may still have written in it.
Indeed, Harry explicitly says that Riddle wrote the diary:
“It was this diary,” said Harry quickly, picking it up and showing it
to Dumbledore. “Riddle wrote it when he was sixteen. . . .
—Harry Potter and the Chamber of Secrets
The purpose of the diary was ultimately to open the Chamber once more:
"Well, he didn't want his hard work to be wasted," said Harry. "He
wanted people to know he was Slytherin's heir, because he couldn't
take credit at the time."
—Harry Potter and the Half-Blood Prince
It is this statement that Dumbledore references when he says that the diary proved that Riddle was Slytherin's heir. Nonetheless, I suspect Riddle originally wrote in the diary about his opening the Chamber of Secrets: it is the only interpretation that makes sense.
It seems most likely that Riddle wrote in his diary the details of how he had opened the Chamber. Thus, the diary proved that Riddle had opened the Chamber before he made it into a Horcrux. After he made it into a Horcrux, the piece of soul inside could control the ink in its pages (and thus presumably kept the account hidden).
Alternately, perhaps Riddle made the diary precisely because he needed a Horcrux that could communicate with people directly, which the diary could do by means of the ink spilled on its pages (whereas presumably the diadem could not). As such, proving that he was the Heir of Slytherin and making the diary in particular into a Horcrux may have been essentially the same goal. The locket did communicate, after a fashion, but only with Ron Weasley, who had perhaps been a bit too emotionally influenced by it.