I merely want to return the name of the context menu relevant to the selection point because of the rich information it provides about the place in the document (much better than range, story etc IMHO).
AFAIK, it does not have a name because it is context sensitive. Say you right click when the Selection is on a field, you get Update Field, Toggle Field Codes etc; if the Selection is on a hyperlink, you get hyperlink stuff, in a blank spot in table, you get table stuff. You know this.

So no, it is not a different name when it is showing the items for being in a table, a different name when it is on a field etc. etc.

It is all the same menu...but context sensitive.

I am VERY curious as to what information you are wanting to get. Could you elaborate?