Now I may well be quite wrong with the current 'recognised' principals around the term 'laying-on' but I want to say how this term came into my fishing back in the late 1950's, and what I perceive is 'the way' to fish the method. Obviously, I realise many of the differing approaches mentioned above are very similar but to my mind 'laying-on' has only ever meant one fixed way of fishing...
Be patient...
As a very young man (teenager actually) I joined several clubs which enabled me to get to rivers such as the Thames along its entire length. Some of the older, very experienced members were always prepared to help me with methods and one of these kindly souls suggested laying-on to catch fish on the bottom particularly with the 14' 'match rod much in use in those days.
Generally speaking, it consisted of two principal items: A longish porcupine quill float fixed top and bottom and a 1/4" dia lead bullet. The float was slid up the line followed by the bullet, and the hook tied on. A stop shot was then applied some distance from the hook (usually) 6/12" and that was it.
The whole lot was cast out a distance, not much more than the end of the rod, and if the float shot under it was move upward until it lay flat on the surface (in other words, overdepth). The rod was then placed in the rod rest and the line tightened until the float just cocked.
The rod then stayed in the rest and the strike was made from the rest in one movement, and not lifted before the strike. Many times the float just disappeared in a flash anyway and the tip occasionally pulled round, so it was very concentrated way of fishing. It worked very well for decent roach and quick bites could sometimes be remedied by moving the shot.
I'm aware this explanation suggests something similar to the lift method and indeed other methods, but my method always included a rolling bullet and it was the only way I ever saw it fished, AND it was only ever known as 'laying-on'. I would agree the rod length/depth of water is the guiding factor in its use, but in (up to 8/9' of water) with good distance between float and bullet ( to avoid line bites) it could mean the difference between ziltch and a few roach in matches.
Not for one moment am I say the term is misused these days and variations to a theme will always crop up, but back some 50 years this was the way it was fished - in my circles!
I still use it today from time to time and find it 'better' than using a Polaris type float or the lift method for that matter - in certain situations such as close in where minimal casting and the bait fixed in one place, it can produce good fish!
ps The size of the bullet was dependent on the flow in running water.....