I read your response to Callie about whether a cone has a vertex or not. Is it ONLY a vertex if both halves of the cone are together or can one half of the illustration have a vertex?
I think the key question is - WHY do you want to use the name 'vertex'?
For (a) or (b) it is clear that if you need to refer to that special point, then you want to call it a vertex. It is special, even when you have half of the figure.
For (c) it is much different. Probably not useful to play around with those words for these objects. There is a context, and a way to use them consistently. That involves working with the flat shapes (the evolutes) which are going to be folded and taped to make a nice 3-D cone. If that really is your context, then you should look at the flat shape, the corners of it, the edges joining those corners etc. When you roll it up and put on the tape, some of those points and curves are no longer special - but they were named and those names should be kept.
We don't usually ask WHY when looking at a definition (or choosing among two definitions). However, mathematicians do ask why, and it is good for teachers and students to reflect on the fact that the names and conventions are a very human creation for very human reasons.
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