They're both mixed together now

Same here

Under the hypothesis that both are canonical at the same time, I would like to know exactly how they relate and reconcile.

Bruce came back to gotham, he fights the red hood gang, he dresses as a veteran and meets catwoman, he bleeds out decides to become a bat, he becomes batman and gets stuck in a building where the police go after him. He helps gordon save his son. Red hood gang is defeated with red hood one falling into vat of acid. Riddler takes over gotham. The bat signal is lit up. They have both happen, but things like Gordon's first time coming from chicago to gotham then in zero year where he's been a cop in gotham for at least 17 years. And also gordon proving his trust to batman in 0 year and batman proving his trust to gordon in year 1. Not everything works, as they should at this point just write a new origin and mix pieces of these stories together or something

Writing a new definitive origin combining these two stories in a coherent way would indeed be the best option and would avoid confusion for readers. I hope they do it someday.

Right now origins are like the thought experiment of Shrödinger’s cat. Both origins are canonical at the same time so long as no one actually writes anything about it. At least Joker Year One takes place between stories.

Really