Re: it's all about the greeting
Mary, funny you say that. When I was a fifth kyu I watched a nidan take her test for fukushidoin (a ranking that recognizes a certain level of teaching ability). Her randori was so graceful as she extended her arm to meet her uke I expected her to sprout opera gloves.
The lesson I took away was not only to feel that I "own" the mat but that I am the hostess greeting my guests and inviting them in.