As to what constitutes an LCC is indeed a difficult one to answer. I'm almost tempted to suggest there are 2 classes of LCCs
My main route in and out of BHX is presently DUS. When I started using that route it was a Lufthansa flight operated by Eurowings. Over the years it has seen some changes and now it is a Eurowings route (often operated by GermanWings). The point is that it progressively transitioned from a legacy type operation to what is now an LCC operation. However, I see Eurowings/Germanwings as a superior LCC model to some of the others out there. It's a personal opinion of course but there it is.