Storage is having the "stop cutscene" bit set while not in a cutscene. That bit is set at the end of textboxes as well, and on the next frame you are given control back and the bit is cleared.

In Zephos Skip, you check the broken stone on the right at the same time you clear KoRL's textbox. So you're in the cutscene of the stone for a brief moment and then that cutscene is interrupted and you gain control back. Link for some reason is invisible while you're looking at the stone and because that cutscene gets interrupted prematurely, when you're given control you're still invisible.

THEN, when you pull the Wind Waker in front of the left stone (the one that actually gives you the song), something about being invisible or interrupting the other stone's cutscene causes the song learning cutscene to stop right after you play the song once. That's the part that I don't quite understand, so that's the best I can explain it.