Grant, I believe the reason it is called 'Southeast' is the highest point is southeast of the White Mountain marker on the 7.5 minute map, not because of the peak's geographic location in the state. This seems a bit extreme in my opinion since the printed name is close to the summit and the mountain should possess the name, So I'll change it to "White Mountain" for the time being.
I got the name info here:
http://peakbagger.com/peak.aspx?pid=2959
When a duplicate name for a separate peak in the state exists, the convention here is to use alphabetical letter increments for each lower peak; i.e. if there are two named "White Mountain", the highest is 'White Mountain A', second highest is "White Mountain B". Once I put all Washington peaks up, I'll append the name with the appropriate letter (looks to be at least "White Mountain D" from a quick Topozone search).