That's Rhaegar with Elia Martell and their son Aegon (the would-be Aegon VI), and Rhaegar saying, "He is the prince who was promised, his is the song of ice and fire." If Jon is Rhaegar and Lyanna's son, then Rhaegar was already dead when he was born (since the theory is that Lyanna died from the strain of childbirth and gave him to Eddard only minutes later before dying herself) and there was no such 'cosy' scene.

To add to the other theories, we know from Oberyn's observations in ASoS that Elia was weak and sickly. A popular fan theory is that Elia was told that if she bore a third child - Rhaegar wanted a third child, a Visenya to add to Rhaenys and Aegon - she would die, so Rhaegar had to seek another mother to bear his third child, possibly even with Elia's consent (the Dornish being fairly relaxed about that sort of thing and Elia apparently being clued in to the prophecy), which led to the whole situation with Lyanna and the war. Where confusion sets in is that Rhaegar was apparently wrong: his third child was supposed to be a daughter, not a son, and that puts the prophecy in doubt (though we already know from the death of Dany and Drogo's son, the stallion who mounts the world etc, that prophecies can be undone in the ASoIaF world).