Yeah, I knew that song too, but not when I was two years old. Jesus Loves Me was probably the first song that I learned, followed by The B-I-B-L-E. . My family was full of Sunday school teachers and choir directors.
When I was about 4, she was with some choir members in our living room, and they made a tape recording of Haldor Lillenas's 1922 hymn "Jesus Will Walk With Me." I became curious as to who this Jesus was who was going to walk with my mother. Later, she told me about Jesus and taught me "Jesus Loves Me."
In 1956, I heard my father play a tape recording of a choir singing a catchy tune which stuck in my mind for the next six decades. I never heard it again until 2006, when I obtained one of my mother's Old Fashioned Revival Hour hymnbooks. The sheet music of an old Methodist hymn entitled "Sound the Battle Cry" seemed to match the melody that had been stuck in my head so long. I surfed over to the old Cyber Hymnal site, and sure enough, that was it--I had recalled the melody note for note.