Rant incoming, so TL;DR: The Painted Worlds are not named after their painters, but the champions who provide their pigment.
I just realized we might have the Painted World concept wrong. When we begin a new painted world, this is what the Painter Girl tells us.
My thanks, Ashen One. With this will I paint a world. Please tell me thy name. I would name this painting after thee.
I've heard that so many times and never quite realized what it was implying. Think about it, we are the champions who bring the Painter Girl flame and blood, the materials needed to create a new world and continue their cycle. She has no name, and so she names the painting after us.
If this is truly a cycle, then it's very likely that Ariandel went through a similar process with the previous Painting Girl. He brought her the materials, and she named the painting after him. However, he is called the "restorer" of the Painted World, which sounds like he rebuilt on top of the old one, but this clearly defies their natural cycle. The inhabitants of the Paintd World burn the old painting away before starting over, as the corvian settler explains:
"When the world rots, we set it afire. For the sake of the next world. It's the one thing we do right, unlike those fools on the outside."
This is their tradition, and Ariandel is breaking it by not letting them burn the world down. So, here's my quick explanation for the scenario:
Ariandel traveled to the Painted World and met his Painting Girl. He becomes the champion that brings her fire and blood, and together, they paint a new world for the forlorn. They become Father and Mother, they have a kid, and everthing's great until Mother dies at some point. Perhaps the effort of painting an entirely new world drains them of their life, who knows? Regardless of how she passed, Ariandel falls into despair, and then Elfreide shows up.
The Sister starts imitating the dead Mother, and bewitches Ariandel with visions of his lost love. She convinces him not to destroy what they created together, and to let it rot rather than be burned away. Ariandel, consumed with grief and loss, refuses to let the cycle continue and locks his own daughter away so that none may show her flame.
Now, this makes the first Painted World even more interesting. With this theory, Ariamis becomes someone like us, probably an undead champion who provided his Painter Girl with the pigment to create a new world. At first, I thought this could be Priscilla, but the game also states she was drawn into the Painted World herself, so that's unlikely. I know Velka has been in every lore theory ever, but she is honestly the best candidate for Ariamis' painter girl, as there is so much evidence of her worship there. However, you could argue that Velka and Priscilla have their own names, so them being Painter Girls doesn't exactly match up.
Then again, I would argue that the painter girls were eventually given names, as we see with yet another crossbreed Yorshka. Furthermore, the community has already named this Painter Girl Aria, so there you have it. Painter Girls are just too cute to go unnamed for very long. It's meta-lore.
This theory's not perfect, but I honestly believe there's some connection between the champions and their painters, which in turn mirrors the champions and Fire Keepers of the "real" world of Dark Souls. Together, both pairs can recreate the world and start a new cycle, but eventually, their partner fades away and their world begins to rot, requiring a new champion to step up and start all over again.