Its a bit complicate, I didn't read the lore version of these two, but for the historical fact, Emden was known for being a beautiful ship and also a very deadly one when the WW1 started. So my guess here is that Emden the white is suppose to be the Emden before the WW1, a more pure version of what she become during the war. But I have no clue if there is another reason explained in the lore of the game, and they was no quotes explaining why she have two personalities.
They represent two different Emdens. One of them is the Dresden-class cruiser that was involved in raids in Asia before being sunk, and the other a Königsburg-class that was named after her, which was scuttled at Scapa flow after the war.
