In the meantime, you can use animated simulators instead :). Some of them:
https://speedcube.site/ (I haven´t tried this one)
Granted, physical cube is physical cube, though.
although I can not help you with the method you provided, I think you might be also interested in the HTA metod or a similar 3-Color method if you prefer the cube as to something to fiddle with.
If you want to drastically reduce the number of moves and use only an intuition, you...
I wouldn't be objecting at all if AbsoRuud added to his/her answer what you have said (sure, it might be obvious for him/her, but I am not convinced it is also so obvious for Shurrik). Especially if a 4x4x4 beginner is asking.
AbsoRuud, by that logic, he/she could solve the dedges/composite edges on the left-hand side simulator (see that earlier linked page) by a U move (plus a 3-cycle of those pieces). However, then he/she would be stuck in an infinite loop (if your recommendation would be applied).
unfortunetely for you, yes, it is possible. Nothing wrong with your cube :-)
If you had a so-called Supercube, it might look as the last simulator in the middle at this site: https://hlavolam.maweb.eu/parity-problem (scroll down to the end of that page).
You can solve your case...
as for a Square-1 notation, you could use the one which basically every speedcuber uses: consider moves of the form (x,y) and moves of the form /. In an (x,y) twist, x indicates the number of 30-degree increments that the top layer should be twisted clockwise (an edge is 30...
apparently it´s been some time since BOCA (dataBase Of Cubes Algorithm) worked perfectly at http://boca.a5.pl.
I would like to ask especially (but not only) Polish cubers: Will it be working again in the future? Or is it even working now somewhere else? Can you give me a contact to...