The part describing the state change is quite logical and easy to understand.
But i have no idea what output is expected or how the given example solves that problem
Is it really
touch LEFT in ROW 0
touch RIGHT in ROW 1
touch LEFT in ROW 2
in a not given order, because if i do that in exactly this order there won’t be all LEDs glowing in the end …
Order doesn’t matter in this puzzle, and yes, those steps actually will solve the given problem. Play with it a bit here:
http://www.ueda.info.waseda.ac.jp/~n-kato/lightsout/
You can create your own puzzle by clicking the Edit button, then you can play it by clicking the Play button.
Try setting it up like:
...
*..
...
Then try touching these spots (in any order):
X..
..X
X..
You will find that it does, indeed, solve the puzzle.
Note. The above link no longer works. You can still find the link in the Waybak Machine, though:
https://web.archive.org/web/20160911054156/http://www.ueda.info.waseda.ac.jp/~n-kato/lightsout/
The EDIT button is obscured by the banner at the top of the page. Click the X to close the banner, and then you can edit the puzzle. This is a good way to test solutions to this puzzle.