Clarifying things always seems like a good idea, but we need to balance it with information clatter. Look how is it done here at StackExchange or Facebook:
Google also solves it this way. Why? Because low of proximity works pretty well. People know what they are editing when they clicking it. However if I was designing for users that aren't good with computers then I will check to be sure if they know what will happen if they click it. I guessing they will know what edit in SE is for (but some of them wouldn't know that they need to click '...' to edit a comment in facebook).
So when we need a full label? When we have a bad, clatter UI. If you look on the examples in the internet, the contextual menu is always near only this one thing that it is about. However adding such clarification is crucial for blind users, although I remember there ware ways to add such clarification for screen reader only.