"To expand on this a bit.

The source of the darkness isn't Lorkhan, it's Namira herself as (per these belief systems) she is the Great Darkness/Void (though the Great Darkness seemingly adopted the identity of Namiira and, in that sense could perhaps be said to have been born, as a result of becoming connected to the just born Lorkhaj) .

According to the Reach version of the tale, Lorkh's connection to the Void/Darkness presumably originates from the covenant he made with Namira, the Spirit Queen, to be granted a place for the creation of the realm of flesh/Mundus within the infinite Void/realm of spirit that is Namira's domain (though no direct confirmation exists).
According to the Khajiiti version (pre or post Riddle'Thar Epiphany), the Great Darkness predates Lorkhaj's birth entirely as it's the place where Fadomai gave birth to him. Because Lorkhaj is born in the Great Darkness, it fills his being and becomes connected to him, thus seemingly either acquiring a sentience and will of it's own, or at least gaining a more concrete one, it was already there but now it has a name as well (And when he was born, the Great Darkness knew its name and it was Namiira).

Either way, the Great Darkness as a force predates Lorkh(aj) in both versions of the tale, it is the place where Lorkh has his epiphany that makes him decide to create Mundus even.
According to the Khajiiti tale (The Favored Daughter of Fadomai), the Dark Heart is created when the darkness inside Lorkhaj attempts to mimic and replace his actual Heart (Heart of Lorkhan), keeping him alive for a time after he suffered a great betrayal (Lorkhan's punishment) but also threatening to have his Void aspects overwhelm him.
Azurah tears out the Dark Heart and casts it away (to stop her brother from being overwhelmed by the darkness), Noctra/Nocturnal is born from the black blood that falls from the Dark Heart as it beats (from the essence of Lorkhaj and Namiira, presumably), Lorkhaj's Void corrupted shade becomes the first of the Dro-m'Athra (The Moon Beast), his true spirit passes into the afterlife (Sovngarde ?) and the Void Heart (described as a reality bending piece of the primal Void trapped on Nirn) somehow ends up in Blackreach (or maybe Blackreach forms around it like with the Heart and Red Mountain ?) to serve as the focus of the schemes of the Dwemer, Nighthollow and, later, the Gray Host.
https://en.uesp.net/wiki/Lore:The_Favored_Daughter_of_Fadomai

Reachfolk recognize only two worlds: the world of flesh and the world of spirit. While Hircine reigns supreme in the world of flesh, Namira, the Spirit Queen, rules over the infinite realm of spirit.

But in the Reach, her dominion extends far beyond mere slugs and darkness. Reachfolk see Namira as the avatar of all primal dualisms. Life and death, beginnings and endings, possibility and entropy—all fundamental competing forces flow from her realm of spirit.

According to Reach myths, Lorkh convinced the Spirit Queen, Namira, to grant him a place in the infinite void where he could create a realm for wayward spirits. Rather than a vibrant paradise, Lorkh created a hard and painful place—a realm that taught through suffering.

It was there in the darkness that Lorkh understood. Nothing does not exist. Where there is nothing there is possibility. And so he found a space in the Void where all that is could be*. —Vateshran Eoinola*
https://en.uesp.net/wiki/Online:The_Dark_Descent

https://en.uesp.net/wiki/Lore:Great_Spirits_of_the_Reach

https://en.uesp.net/wiki/Lore:Spirits_of_Amun-dro

Namiira. The Eldest Spirit. The Great Darkness. The Void. All creatures who feed on rotten flesh are her spies and the prey of Cats. The Lunar Lattice protects us from her hunger, but not our own. Know that to name her aloud is to invite the Dark, so you must never do so, as Namiira is the sound of her true name. She is a spirit of infinite realms, of which only Azurah knows all. Mortals who become ensnared by this spirit are tortured until they forget who they were and know only Namiira. This is eternal suffering for all souls but the ja-Kha'jay, whom Azurah will not abandon to the Dark.

Lorkhaj. The Moon Prince. Fadomai's Favored Son. The White Lion. He was born in the Great Darkness and it followed him as his burden. Loved by many, he was considered a noble leader. Lorkhaj was the first spirit to make his own path with purpose, because he was in conflict with himself as soon as he was born. His courage inspired all those he encountered, so much that he united the spirits to make the World. He gave his life to do this. We honor his sacrifice by walking the Path with purpose and resisting the call of the Dark. Lorkhaj represents the duality of the Khajiiti soul and the hardships that all Khajiit must overcome. In her wisdom, Azurah lit her brother's pyre with the Twin Lanterns of Jone and Jode, and thus the true spirit of Lorkhaj will sometimes appear—but only when called by Azurah or Khenarthi, or by his oldest name.

Lorkhaj. The Moon Beast. Born of the dark heart of Lorkhaj, which overcame him after he suffered a great betrayal. In his wisdom, Lorkhaj sought his sister Azurah, who tore the darkness from him before it consumed him and cast it into the Void. We know this shade of Lorkhaj as the first of the Dro-m'Athra, who serve the Ur-Dra Namiira, our enemy. The Moon Beast prowls the edges of the Lattice and pounces on Khajiit who stray too far off the Path. Know that on nights of the Ghost Moon, Azurah has opened the Void Gate and the Moon Beast will challenge mortals until banished. We accept this burden as part of the Path, and for the sake of our Lost Kin.

Ahnurr caught Fadomai while she was still birthing, and he was angry. Ahnurr struck Fadomai and she fled to birth the last of her litter far away in the Great Darkness. Fadomai's children heard what had happened, and they all came to be with her and protect her from Ahnurr's anger.

And Fadomai gave birth to Lorkhaj, the last of her litter, in the Great Darkness. And the Heart of Lorkhaj was filled with the Great Darkness. And when he was born, the Great Darkness knew its name and it was Namiira.

https://en.uesp.net/wiki/Lore:Spirits_of_Amun-dro

https://en.uesp.net/wiki/Lore:Words_of_Clan_Mother_Ahnissi

https://en.uesp.net/wiki/Lore:Dark_Heart

https://en.uesp.net/wiki/Lore:Heart_of_Lorkhan

https://en.uesp.net/wiki/Lore:Dro-m'Athra"