After yet more random trial-and-error testing, I have determined that the field order in the attribute table is always based on an alphabetical sort on the field alias. Thus, if my XML looks like this:
<?xml version="1.0" ?>
<configuration>
<layersettings>
<layer name="Surface USFWS"> <!-- type="feature" -->
<columnsorder>fields</columnsorder>
<fields>
<field name="LRS_Tract" alias="LRS Tract"/>
<field name="Refuge" alias="Refuge"/>
<field name="Notes" alias="Notes"/>
<field name="NWR_Unit" alias="Unit"/>
</fields>
</layer>
</layersettings>
</configuration>
Then the attribute table looks like this:
[ATTACH=CONFIG]23110[/ATTACH]
However, if I add letters in front of the field alias names, then I can force the fields to sort in the order which I want to see in the attribute table. But this is still regardless of the order that the fields are listed in the XML. While this method achieves the desired results, it is still unacceptable because now the field names have a, b, c, d, etc. appended to the front of them, which is silly and doesn't make sense from an end-user standpoint. So this code:
<?xml version="1.0" ?>
<configuration>
<layersettings>
<layer name="Surface USFWS"> <!-- type="feature" -->
<columnsorder>fields</columnsorder>
<fields>
<field name="LRS_Tract" alias="c - LRS Tract"/>
<field name="Refuge" alias="a - Refuge"/>
<field name="Notes" alias="d - Notes"/>
<field name="NWR_Unit" alias="b - Unit"/>
</fields>
</layer>
</layersettings>
</configuration>
Results in an attribute table that looks like this:
[ATTACH=CONFIG]23111[/ATTACH]
So, regardless of what ESRI states in the "What's New for 3.2", the attribute table widget does not honor the field order as specfied in the AttributeTableWidget.xml file.
P.S. I have officially submitted this as a bug.