The OP is right! Look in the Mishna Brura he mentioned. The Mishna Brurah brings in the name of the Shelah that a woman should not sing a baby to sleep with a non-jewish MELODY, because it can have an affect on the baby’s neshoma.
Music is a very sublime form of expression, and a person’s music is a kind of product of his soul (it says so in the Kuzari). Therefore, even music without words can have an effect on you.
This is true even though the baby cannot understand one word, even if the mother is singing with the words.
Even if you hardly pay attention to the words, that, too, won’t matter. The words are still there, you’re still listening. They still get into your head, and stay there. It is known that even while sleeping, people can listen to instructive tapes and it will make an impression on their minds. To wit: Lack of attention does not negate the effect of the words.