Cooked omelet and cooking Japanese omelette is another thing. Most omelettes are made with egg only while there are various styles and types of tamago:http://mesubim.com/2013/04/14/tamago-kizushi/

This is tamago served as a food and it has some sugar or miring and or sake or dashi added. Then served with daikon grated and some neggi, the classical combination. The tamago is always made with a Tamagoyakinabe or tamagoyakiki, a specific pan either square or rectangular shaped. The egg is rolled in the pan and the cooking is made in steps: http://mesubim.com/2013/04/15/tamago-yaki/

The idea of tamago in sushi is all about the style of the chef, his own recipe and the same way that tsume is the chef’s homemade tare, or his nikiri that he swipes over his fish. In fact in sushi there are only a few things that chef’s cook and the rest is raw. So these are the distinguishing factors when it comes to sushi chefs own identity.