Yeah, that all sounds very similar. Once I figured out how to move in water, I tended to do what your daughter does, swim under the water and surface for air. I still like swimming that way.
It's interesting that people I know who went through a very formal swimming program as youths and then think of swimming as exercise and laps, tend to struggle with below-the-surface activities, like diving for items.
I remember my brother had a whole slew of underwater games we used to play like "tea time", "hold your breath at the bottom the longest", "headstand", "underwater flips", "drag your friend while they're underwater while you run in a race", "pull a gallon of dirt off the bottom", "upside-down Buddha" and so on including some pretty intense and dangerous-ish horseplay. There's probably a more comprehensive list somewhere as he didn't invent all of these but learned them elsewhere. Looking back I realize that we learned breath control, buoyancy control, how to handle strange orientations and direction underwater, diving, getting accustomed to handling stale air or out of air situations. But it all happened naturally through play.
Growing up, the only other kids I knew who could do the same kind of things also learned in similar informal ways built around play.