When is a brand born?

If brand equity is defined as:

"A set of assets and liabilities linked to a brand’s name and symbol that adds to (or subtracts from) the perceived value of the product"

When then is a brand considered born?

Answer:
A brand is born the day the federal government recognizes a registered trademark for it.

Before that happens, it's just a concept or, worse, a label for a good for which the label is "unprotected" against being copied.
Don't confuse a trademark (legal registration), which does not have almost any value unless is also a brand, with a brand name.
It is born the moment that the public starts identifying the product with a name. That is the brand.
According to your definition, which is right, they start associating assets and liabilities to that brand name. The brand is born.