Chronic, realistically, you are correct, but it makes more sense to me to count from the flip because at that point you are actively transitioning. Keeps in more organized in my opinion.
I don’t think it’s realistic to call the very first day of 12-12 the first day of flower. It does take the plant some time to adjust. If you were growing in natural sunlight you would watch for pistols and start counting. I usually count from when the first white pistols appear, which could be as much as 4-7 days after the switch depending on the strain.
Flower starts when you first see the white hairs (stigma). After that, it's 2-3 weeks of preflower, then full bloom.
During preflower, you'll want to slowly take put the N nutes and add the PK nutes.