Sappho

It contains the temple of Apollo Leucatas, and also the "Leap," which was believed to put an end to the longings of love."Where Sappho is said to have been the first,
"as Menander says,"when through frantic longing she was chasing the haughty Phaon, to fling herself with a leap from the far-seen rock, calling upon thee in prayer, O lord and master.
"Now although Menander says that Sappho was the first to take the leap, yet those who are better versed than he in antiquities say that it was Cephalus, who was in love with Pterelas the son of Deïoneus.

Sappho was the famous poetres from Lesbos who jumped the rocks of Leucata to be freed from here love for Phaon around 580-570BCE.

Geographica 10.2.9 Strabo

Theodore Chasseriau, Sappho Leaping into
the Sea from the Leucadian Promontory

Ptolemy Hephaestion listed the following people who did jump the rock of Leucata:

Aphrodite for her love for Adonis: freed.
Artemisa for her love for Darnados of Abydos: killed.
Hippomedon of Epidamos for a young boy: killed.
Nikostratos for Tetigidaia of Mirina: freed.
Makes of Buthroton jumped 4 times: freed.
Boulagoras of Phanagorite for Diodoros: killed.
Rhodope of Amisene for Antiphon and Kyros: killed.
Nireus of Katana for Athena: freed.
And the poet Kharinos for his love for the eunuch Eros, butler of Eupatos. He broke his leg and died of pain and wrote this poem:
"To the devil with you,
deceptive and murderous rock of Leukas!
Kharinos, alas! alas! this jambic muse,
you have turned by to cinders
by your vane words of hope.
Can Eupatos suffer so much for Eros."

So yes, Odysseus made an impression to take the leap first driven by love. This is the reason why these cliffs of Leucatas became famous by lovers who took the leap here also out of love.

So, there was discussion about who jumped first, Cephalos was born around 430BCE. We now know that Odysseus was the first to jump for his longing.