One thing that confuses people about Microsoft Project is that there is often more than one way to do things. Naming fields is one of those things. As far as I am concerned there is a right way and a wrong way.
This is the wrong way:
Why is it wrong? Well, the change is only made in that place. There is a new title for the field, but the title is not as widely used as the name. To really rename a field so you can use the new name in grouping, filtering, reports etc. you need to use a different method.
Here is the correct way to rename a field:
Now any dialog boxes that present a list of fields will show the new name in the list - the old one will too so don't worry.
Changing the field name applies to any table and any view that uses that field. Changing the field title (the first method)is limited to the single table that you have changed.
My guess is that when project first came out you used the first method of changing the field title, but after a while they improved the functionality. However, they never cleaned house and removed the old method. In a way this is understandable - Why fix what is not broken? But on the other hand, eventually you end up with so much old stuff that the user interface becomes cramped and confusing. Project has definitely reached that point.
One more thing to discuss and that is Master Projects and Inserted Projects. If you have changed the field name in a sub project and then inserted it into a master project the new field name will not show up. You MUST change it in the master project for the change to show. This is also true for views and other sorts of formatting. The parent does not inherit from the child.
Note to those who prefer the old way: Yes I can think of some situations where changing the field title instead of the field name would be useful, but I'd be willing to live without the function.