This is very close to the lullaby my mother sang to all of us kids, and we have sang to the little ones who follow us.

The movie "Awakenings" uses this song.

The words are a bit different, and we never knew (until "Awakenings") that there was a second verse. I've never known anyone who knew this lullaby.

The way we learned Baby Boats:

Baby's boat a silver moon, sailing near and far.
Sailing over a sea of blue, while the clouds roll by.
Sail, Baby, sail out across the sea,
Only don't forget to sail home again to me.