Mythological inaccuracy

Just at the beginning of the movie, the guide informs visitors:

"It was here, on the Acropolis hill, that some believe Theseus's father, King Aegeus, looked out to sea, waiting for his son to return.
See, Theseus had promised his father that if he defeated the Minotaur,
he'd hoist a white sail on his ship as a tribute but he was so excited
to get back home he forgot and he flew the same black sail that he set out with.
Well, thinking his son was dead, the heartbroken king flung himself onto the rocks below,another victim of the cruel tricks gods play on men."

However, this is supposed to have happened not at the Parthenon,but at Sounion Cape, some 60 km south of Athens.