Priests of R'hllor really suck at the whole prophecy thing compared to your average village witch like Maggie the Frog. She nailed that shit within seconds of sucking on a drop of blood.
Rhaegar was trying to spawn the prince that was promised but damn near everything points to his sister being the Neo. It's the "three heads" parts that is fucking with everyone, which could simply be referring to the 3 actual dragons and not their riders.
And Mel thinks this prince is also Azor Ahai reborn, which is probably just R'hllor's shitty brand of foretelling confusing everyone and they are two different people. But that's the mega obvious Jon + Dany theory that everyone dismisses as TOO obvious.
Prophecy is just a tool for GRRM to spend 20+ years fucking with us. Even on a re-read, most of the shit shown to us in visions or prophecies remains vague, aside from on-the-nose shit like Dany witnessing the Red Wedding.