I'm not sure if this is true or not, but the same company that makes these dolls also makes Moxie Girlz. One of the Moxie dolls was sold with a "darling rag doll" as a toy. It looks just like a tiny Crumbs Sugar Cookie. So that little doll may have been the first LaLaLoopsy Bitty Buttons doll.
I don't know how much time has passed between that little doll's creation and the bigger dolls, but they both were probably designed by the same artist. This is a guess on my part. I could be completely wrong in that assumption, but that's my guess.
I bought that particular Moxie doll on ebay JUST to get the "Darling Rag doll!" When she arrives, I will post pics of her. Then I will sell or donate the Moxie doll, because I don't collect those.
I am digging up info all the time, but some of it may be bogus. I did find out about the reasons for the name change and they are posted in the links at teh bottom.