I think it makes far more sense that Rin died on the same day and year the Kushina was giving birth because it would be far weirder that their kid's birthdate happened to be exactly the same day in a different year.
Also from the backstory we know that Minato and Kushina were moved to a secured location away from the village so it's likely that he was unaware of the situation involving Rin and Kakashi until after the fact, and explains why Obito was wondering where the 4th was (offsite preparing for the birth).
Also from the backstory we know that Minato and Kushina were moved to a secured location away from the village so it's likely that he was unaware of the situation involving Rin and Kakashi until after the fact, and explains why Obito was wondering where the 4th was (offsite preparing for the birth).