Oracle Text CTXCAT Domain Index, Transactional, and Before Insert/Update Trigger
Aha!  I've found the answer.  Talk about an edge case.

First, I found this post from 2007, where someone says:


  ...the AFTER trigger for synchronizing the CTXCAT index on [column] is
  not firing (since my update statement does not include the indexed
  column).
  
  ... 
  
  Oracle, would it not be better to generate the CTXCAT
  trigger to examine the :old and :new values in the indexed column,
  rather than using a check on (if updating([column]))?


Over a year later, someone replied:


  I fixed this issue by adding additional UPDATING (first_name and
  last_name) conditions in the DR$table_name trigger.
  
  Here is part of the modified trigger. 
  
  if (inserting or updating('LAST_NAME_FIRST') or updating('FIRST_NAME')
  or updating('LAST_NAME')) then reindex := TRUE;


Hm, so the CTXCAT index uses a trigger to know when it needs to update the index for a particular entry.  All I need to do is tweak the trigger and recompile it so it does what I want.

To get the content of the trigger:

SELECT text
  FROM   user_source
  WHERE  name = 'DR$NAMES_IDXTC'
  AND    type = 'TRIGGER'
  ORDER  BY line;


I copied this into Sublime, prettified it, and found this (simplified):

TRIGGER "TEST"."DR$NAMES_IDXTC" AFTER
INSERT
OR
UPDATE ON "TEST"."NAMES"
FOR EACH ROW DECLARE REINDEX boolean := FALSE;

BEGIN 

IF (inserting
    OR updating('COMPOUND_NAME')
    OR :new."COMPOUND_NAME" &lt;&gt; :old."COMPOUND_NAME") THEN REINDEX := TRUE;
END IF;

...

END;


You can see that the 12c Oracle Text version's AFTER trigger does actually compare the :new and :old values of the indexed column to see if it needs to update--not the case back in 2008.

So...if I'm updating the :new value in my BEFORE trigger, that should be reflected in the AFTER trigger, and the comparison would kick off an update to the index.  What gives?

Well, here are the two SQL statements I was using:

update test.names set 
  first_name = 'Skye', 
  last_name = 'Fillingim'
  where ... ;

update test.names set
  first_name = null,
  last_name = null
  where ... ;


(Slightly different than what I said in my question, I apologize.)

The effect here is that, each time I used one of these statements, either the :old.compound_name or :new.compound_name would be null.  So when we get to this condition:

OR :new."COMPOUND_NAME" &lt;&gt; :old."COMPOUND_NAME"


We are doing an equality comparison against a null, which evaluates to UNKNOWN.  Hence, no index update.

This is actually an extreme edge case, because you have to be indirectly modifying :new.column via a trigger, and either :new or :old must be null.  I would never have discovered it it I hadn't used those exact SQL statements.

So, we have a slight update to the trigger:

IF (inserting
    OR updating('COMPOUND_NAME')
    OR :new."COMPOUND_NAME" &lt;&gt; :old."COMPOUND_NAME"
    OR (:new."COMPOUND_NAME" IS NULL AND :old."COMPOUND_NAME" IS NOT NULL)
    OR (:new."COMPOUND_NAME" IS NOT NULL AND :old."COMPOUND_NAME" IS NULL)
    ) THEN REINDEX := TRUE;


And then just stick CREATE OR REPLACE in front of the old trigger, recompile it, and everything works perfectly.