Laydown...
...just means that, when in use, the insert lies more or less flat in the holder. In other words the screw that secures the insert to the holder passes through the hole in the vertical dimension.
The counter part insert is the "vertical" insert where the screw is horizontal and the insert stands on edge.
I use the laydown system and have found that the full profile inserts produce a very nice finish without having to resort to anvils under the insert. The partial profile inserts that cut a range of threads work well for finer threads but not so good for the coarse threads found on barrel tenons.
As you can see on that one graph 16-18 TPI on work a little over 1 inch in diameter is getting out of the ideal envelope of the helix angle established by the holder pocket.
With a full profile insert I can get beautiful finishes in stainless barrel steel at speeds as low as 45 rpm. I use dark cutting oil, Texaco Sultex D, but for this I think any dark thread cutting oil would work fine.
HTH