It seems like you want to add a Split Button, a button with alternative actions to the default action.
Visual Studio does a similar thing for the action "Add Existing Item" to a project. Pressing the button itself performs the default action (here: "Add"); but pressing the small arrow button shows the alternatives to that action:
In this example, the down arrow is actually a separate button, so it requires a click exactly there to open the drop-down. It seems like you want to make the entire button expand the dropdown in your case, but if the requirements does not mandate exactly that behaviour, I would rather choose the Visual Studio way if "Save" would be much more frequently used than "Save As".
You refer to the StackExchange button on top of this page, but that button is not a default action with alternatives in a drop down, but instead expanding a content that is very different from the button itself.
Either way, it is OK to have the label of the button to only describe the default action. Also consider different icons (or no icon for the alternative).
My experiences from the Visual Studio button is that people have a hard time figuring out how to find the action "Add As Link" to start with. They look for something like "Add Existing Item As Link" next to the "Add Existing Item" action in the menu, not realizing that they first have to open the "Add Existing Item" first.
But in your case, with the button with the alternatives already in the tool bar, it will be easy enough to find.