So, what happened to Ornstein?

For me personally, the question what happened to Ornstein has always been one of the most interesting ones in Soulsborne. But the three Dark Souls entries are very vague when it comes to Ornstein.

\- In DS1, it is suggested that Ornstein dies in Anor Londo.

\- In DS2, we encounter the Old Dragonslayer in Heide.

\- In DS3, it is implied that Ornstein left Anor Londo in search of the Nameless King.

So what happened to Ornstein? Did he really leave Anor Londo behind to search the Nameless King, and did he then die at his hands? Did he never leave and someone else stole his armour after the death of him and Smough, pretending to be him in the Cathedral of Blue? Or was the Old Dragonslayer in fact Ornstein?

Is there any theory that could provide an explanation that makes the three games not contradict themselves? I've been scratching my head to find a way to explain this but I just couldnt figure anything out