Outline on text acting unexpectedly

C,

I for one hope you will share your findings.

And I hope you will notice that Hans-Jürgen also added the Drop Shadow to the bottommost Fill to make everything complete in the exact right way.

Claire,

Do can do as follows with one single editable instance of live Type (editable texdt) or one single instance of  outlined Type (paths):

1) In the Appearance palette create New Fill twice and Drag both of the new Fills down below Characters/the original filled path;

2) Select the top new Fill and Effect>Path>Offset Path by the desired thickness of the inside stroke and set the colour as desired;

3) Select the bottom new Fill and Effect>Path>Offset Path by the desired thickness of the inside stroke plus the desired thickness of the outside stroke and set the colour as desired.

"I'm going to feel stupid "...

I hope not, because you only get smarter by asking questions.  I like to try to help people, so here it goes.  The original text, when converted to outlines, is automatically "grouped".  So, it could be assumed that when you deleted the Black ( or Red ) outline, you deleted that "group" ( hence, it would not still be there ).  The existing White stroked outline was / is still there and is probably a copy of the original outlined text ( or still could be the orinal outlined text with a White stroke and a Black fill ).  The originator could have used three copies: 1.) original text outlined with Black fill, 2.)  copy of original with a White fill, 3. ) copy with Black fill and Black stroke.  Or a mixture of orinal text and copies.  I think the entire scheme could be created with the original text and just one copy.  Then again, there may be something else going on.  Best practice is to contact the originator and see if 1.) you can get a file with "live" text, or 2.) ask them what font they used and buy your own version to use on derivatives.  In any case, you should proceed with permission to edit the file per originator's copyright protections.  Many people do not realize that edits are copyright infringement unless they get permission from the originator ( which should not be hard to do ).