No, based on the current documentation, the value of the Name tag is supposed to be the name of a dataframe; not the active dataframe. It is true that when inserted from the ArcMap menu, it automatically takes the name of the active dataframe (which is logical). And in retrospect, it may be a good thing that it shows the name of the active dataframe although a warning should also be given.
Where there is a problem is when ArcMap doesn't update the value of the name TAG when the dataframe of that name has been removed. Ideally, it should warn the user that the tag no longer corresponds to an existing dataframe and inform that the value of tag will be set to a reserved keyword to reflect that it will now on display the name of the active dataframe, for ex. ArcMap_ActiveDataFrame (or whatever keyword that is unlikely going to be used to name a real-live dataframe). Currently the TAG is left to the name of the removed dataframe.
At the very least the behavior that it takes when the value of the Name tag is no longer correct should be documented.
The application I am programming starts with the value of the name tag and then tries to find the dataframe of that name. The way things stand now, that dataframe may not exist! That should be documented; it would have saved me a couple of days trying to figure out what was wrong in my complex program... while it was matching Esri's documentation and was logical.