‘Lining the switch’ is like ‘throwing the points’, and both relate to the moving blades. Indeed, some British companies used “switch” and others used “points”.
A turnout is a combination of points/switches and crossing [vee]. Sometimes this is called a “single lead” in the U.K., especially by permanent way people.
Oh yes, permanent way is the same, but whereas we use the term to encompass what it is and its maintenance, in North America they are called Maintenance of Way, or MoW.
Turnout numbers refer to the crossing angle. In North America these are frequently cast units including the guard (wing) rails and correctly referred to as frogs. Switches are referred to by their length, just as they used to be over here, but things tend to be more standardised and a 16’6” switch seems quite common. (I am aware that this is a simplification.)
Peco code 83 and their new code 70 match American practice, using turnout numbers. The code 100 and code 75 ranges use a constant crossing angle with varying radius (large, medium, small) on the curved road.
Although modellers (I suppose I should say “modelers”!) might refer to the radius of a curve, prototype railways use degrees of curvature degrees of curvature, which is the angle subtended by the curve over an American chain, which is 100’, unlike the British chain which is 22 yards (or 66 feet - the distance between cricket stumps).
Turnout timbers are also ties, and those either side of the switch rod (stretcher bar) extend some distance, about 8’, beyond the other ties to accommodate the switchstand, which is used to line (or indeed throw) the switch.
Because of the distances involved, and the open nature of the landscape, operation outside of densely populated areas tends towards fewer but longer trains. In suburban areas, things are closer to U.K. practice. Understanding this is the key to understanding the differences between European and American practice, and it starts to make more sense.
The different requirements and different terminology can lead to confusion, but if you step back, take a breath and spend some time reading up on things, you soon get the hang of it.