Document Libraries allow you to specify custom template files. Using these - when you create a document from the template it creates a new one and saves it directly into the library. They way you do it will almost always lead to the same result because by default when you open the document from SharePoint it want's to save back there to the same file.
To try this - do the following:
- Create a new document library
- Open the library and in Ribbon select the "Library" tab, then open "List Settings"
- Select the "Advanced Settings" link
- Now - you should see the "Document Template" section on the page. you can point to a document in other library by putting the URL here. Of use the "edit template" link that will open a blank word document that you can edit and then save it back to the library.
- Once you have template defined - go back to the document library view, in the Ribbon select "File" and then hit the drop down arrow under "new document" and you should see your template here.
- Click that - it will open your template - then when you save it - it will still save to the library but it won't overwrite your template.
You can add multiple templates - but to do so you need to use SharePoint Designer or Open the library in Windows Explorer. You just need to store them in the "forms" folder which is hidden by default.
EDIT
If you want to keep the templates in place - then add them all to a library, and don't give anyone edit permissions (just read only). Then they can't save back to that library and overwrite your template. However I believe my suggestion above will be a bit more user friendly.