The "rainbow" version comes from later generations of girls having no idea what a "rain barrel" is. I sang it with "rainbow" when I was a girl in the 60's. My mother tried to explain to me how it should be, but I didn't buy it. After I grew up, of course, I understood what she was talking about. I even tried to convince a friend's daughters that it should be "rain barrel" but they wouldn't hear if it either!