Of course it helps. Any attempt at identification helps.
I never did learn the real name of the rose because J&P never did tell anyone the registration name. Mine has never produced any blooms that weren't pale pink with a peachy-pink center.
It's possible, of course, that your mother's rose and mine are not the same. Maybe J&P never did have an actual rose named Caribbean Breeze and they simply sent out whatever they had in surplus to fill the orders for it. I remember reading a post long ago by a former Michigan Bulb emplyee, who said they never really had a rose called 'Nearly Black,' but the name appealed to a lot of people, so it was one of their biggest sellers. They would send 'Oklahoma' to the customers instead. If they ran out of 'Oklahoma,' they would send another dark red hybrid tea.
I think you're confusing Miami Moon with some other rose. It isn't just various shades of pink. You can see the color variations in the photos in our database.
Rose (Rosa 'Miami Moon')