Sometimes, in the interest of avoiding confusion, these designations are necessary. As a longarm quilter, when I post a photo of a quilted top I did not make but I quilted, it is easiest refer to the maker of the top the "piecer" and myself as the quilter. Since those are the literal definitions of each part we took in the process.
As far as the piecing being the most important part, I don't agree with that sentiment. I feel each part is equally important. Without the quilting there is no functionality in a completed top. It can't be used until quilting (or tying) and binding is completed or it is finished in some other fashion be it a duvet, or simply backed with another piece of cloth and "birthed". I have seen many a quilter turn a poorly pieced top into a thing of beauty and in some cases quilting may be the only thing that will hold all that patchwork together after rigorous use. Additionally some of the most breathtaking quilts I have ever laid eyes on aren't pieced at all, but wholecloth quilts.
Funny how we use these terms. When I start a new project I say I am making a quilt, even if I am at the cutting stage. I don't say I am making a top. However, that pile of tops waiting to be quilted I call tops not quilts in waiting. LOL.