Indeed installing the Integrated Installer is different than flashing the board. The integrated installer installs Edison's drivers as well as the desired IDEs on your computer, so, this process does not interact directly with Edison but rather it makes sure the host computer will be able to work with Edison.
In order to flash the image on Edison you have to use Flash Tool Lite. You can find this tool and a guide on how to use it on Edison Downloads. There used to be a way to flash your board by putting the image files on Edison's drive but it is no longer used since the latest image is actually bigger than Edison's drive, that's why your Edison's drive is empty. You might also be interested in reading Flashing the Edison Module.
Let me know if you have any other doubts.