At what time is a witch/wizard categorized as one?

[Inge]

No: HPFGUIDX 88798

I wondered if Petunia ever suspected Dudley to be a wizard.
When Dudley was first born - and during his first years of growing 
up - there was no way Petunia could know if maybe he would turn out 
to be one.
Even Vernon may have had his thoughts seeing how he knew that two 
Muggles are actually able to produce a wizard/witch (Lily's parents).

Wondering also when one can know for sure if one's child is a 
wizard/witch or not. Is it very early in the childs life or isn't it 
until the letter from Hogwarts arrives?

In Harry's case it seems it was known wizardwide that he was actually 
born to be a wizard since his name had been in the books at Hogwarts 
since his birth.
But what about Neville? Didn't his grandmother suspect him to be a 
squib?
And Hermione? Was she in the books of Hogwarts since birth or didn't 
her name show up till she turned 11?
So when does it come clear that a wizard is a wizard?

Sorry if these things have been discussed to death - I've been away 
from this group for quite a while.

Inge