Though I don't have the reference with me, the story is a little different (it may be apocryphal anyway). The story goes that Lincoln asked the band to play "Dixie" because it was a favourite song of his, and he hadn't been able to hear it for so long. He is reported to have said that it could be considered one of the captured "spoils of war". It was certainly not to honour the Southern soldiers.
yours, Peter T. (I don't know the other song)