Apparently the definition is given in a paper by J. Mercer from 1908, but I haven't been able to locate that paper yet. I will probably just use the terminology since it does not appear much at all in a Google search.

I haven't actually looked at the paper yet except for the part you provided the link to. I was really wondering where else it might have been used and/or defined ?

OK, thanks Anton. I'll look at it more closely to convince myself.

That appears to be a nice way to prove what I was looking for, though I want to look it over more for now to make sure it does not implicitly assume anywhere what one wants to prove. Is the statement that the boundary contains at least $[\dfrac{n+1}{2}]$ arcs of $C_1$ just based on the fact that $C_1$ is tangent at the $n$ sides of D ?