I've been through this so many times when I haven't put my middle name on my booking, but in the end nobody cares but me. I remember being on hold for an hour trying to get Emirates for instance to put my middle name in, and they kept saying their system isn't designed to have middle names and it will auto correct them to one name. I called Delta too and made NUMEROUS calls to Aeromexico, etc. The answer is, it's not important. But, I make sure I do it now for "me" for a peace of mine. After the fact can be a pain.