So reading Microsoft's KnowledegeBase article on this gives the infuriating answer that this is by design, so you need to write a macro to do this. Handily they provide a Macro which updates the fields when you open the file, rather than save. This actually makes more sense for a number of fields, so as long as you use LASTSAVEDBY and SAVEDATE these will make sense.
Just remember that you won't see the updates applied until you open the document again. For footers this is absolutely fine.
And since this is a Macro you definitely need to save as dotm.
For anyone who needs to update fields on save, there is a discussion at VBA Express forums, or you could just adapt the macro from Microsoft's document.
Update
The vbscript from Microsoft is good, but results in a document that always thinks it has been edited, so always prompts to save when closing. Most users will always click on this, which would result in the LASTSAVEDBY and SAVEDATE values showing when the doc was last read, and who by.
So I added a line to immediately mark the document as unchanged after updating the fields. Any further edits to the document will reset this, giving the save prompt on close.
Here's the macro code:
Sub AutoOpen()
'
' AutoOpen Macro
'
'
Dim aStory As Range
Dim aField As Field
For Each aStory In ActiveDocument.StoryRanges
For Each aField In aStory.Fields
aField.Update
Next aField
Next aStory
' set document as unchanged (prevents save dialog popping up when closing)
' further edits will set this back to false and restore
' the save dialog on close
ActiveDocument.Saved = True
End Sub
Update
The above code works fine, unless you save this to your Normal.dotm template, in which case it runs everytime you open any document. If the document is opened in Protected View (i.e. it was downloaded from the internet or came as an email attachment) then ActiveDocument isn't available. The solution is to check that the document isn't in Protected View before running the functions inside the macro. Here is the updated macro code, including the If statement to check the doc isn't in Protected Mode:
Sub AutoOpen()
'
' AutoOpen Macro
'
'
Dim aStory As Range
Dim aField As Field
' Check that document is not in Protected View before doing anything
If Application.ActiveProtectedViewWindow Is Nothing Then
For Each aStory In ActiveDocument.StoryRanges
For Each aField In aStory.Fields
aField.Update
Next aField
Next aStory
' set document as unchanged (prevents save dialog popping up when
'closing) - further changes will set this back
ActiveDocument.Saved = True
End If
End Sub
Create a new, blank document using the template. Copy & paste from the existing document into the new one. I am sure that Word 2013 has something similar to Word 2007's setting that is reached with the start button, Word Options, Advanced (from the left in the tree), go to the "Cut, Copy and Paste" section, and make sure that "Pasting Between Documents" is set to "Match Destination Formating". You will still likely have to select any indexed items (i.e. for table-of-contents) and apply the appropriate style, to have the index work, and then select to update the TOC or any other indices.
Best Answer
You should Create a site Content Type. This allows you to have your predefined templates built in to SharePoint Document Libraries and then use the New button in a given document library to select from the available templates. This is much better then your current approach.