living soil
I dislike this term, since if you were to go dig in a native grassland, one of the most nutrient rich and biologically diverse soil ecosystem out there, it wouldn't meet the definition they're getting at here.
The system they are describing is manmade one with heavy organic inputs. Nothing wrong with that, just semantics I guess. It's kind of disheartening that horticulture is so far removed from growing things in actual soil that we need to specify it when compared to other growth media.
One other thing I want to add is that not all soil amendments are created equal, and some may have both positive and potentially negative effects. A good example of this is cattle manure. They have good C:N ratios, and supply nutieents, but EC and SAR (salinity parameters) can be high. It's in your best interest to test both fertility and a bunch of other things before applying them