"Complex confluence of events, of energies, of circumstances" seems to me to pretty much rule out it being the date alone.
(Always presuming Lash isn't lying.)
As I said in Reply #5, I associate Lash's description of how Harry gained the "potential to wield power over outsiders" with Bob's description of a "conjunction"
"Right then!" Bob said. "The only way to kill an immortal is at certain specific places."
"And you know one? Where?'
"Hah, already you're making a human assumption. There are more than three dimensions, Harry. Not all places are in space. Some of them are places in time. They're called conjunctions."
"I know about conjunctions, Bob," I said, annoyed. "When stars and planets align. You can use them to support heavy-duty magic sometimes."
"That's one way to measure a conjunction." said the skull. "But stars and planets are ultimately just measuring stakes used to describe a position in time. And that's one way to use a conjunction, but they do other things, too."
To me there is a lot of "you're a mortal that can hardly encompass this concept" handwaving going on here that makes me think that if my hunch that the "conjunction" idea and the "complex confluence of events, of energies, of circumstances" are related in some way, then the "starborn conjunction" might have been about something more than just a date.
By the way, I highlighted the "when stars and planets align" line of that quote, because maybe a starborn conjunction involves a
period of time like when "Saturn is in the constellation Badger and Jupiter is in
'Retrograde Motion'" or some such.