Basically, you are probably safe no money changes hands between the performer(s) and the family of the person who is celebrating a birthday. I say probably. There can still be an issue if the situation is public enough, although (as mentioned above) this rarely happens.
But the song's owners can still make trouble for you if they want. Whether they win in court is another matter.
And you conveniently left out the portion that prohibits singing it in a public forum which is defined.
You are correct that the songs owners can sue as they please