Sherpa makes a bag especially for Delta . . . here's a link:
http://www.sherpapet.net/products/de...ddetail_post=Y
This bag is only 10.5" tall. I think the woman gave you the dimensions for a hard-sided carrier in the event you "check your pup as cargo" unfortunately.
My daughter's gym bag has a difficult time fitting under some airline seats. I would go with 10" or less, but that is my persional opinion based on experience.