Is "layer" simply an abbreviation for "feature layer"? Is there any instance where a layer would not be a feature layer, or vice versa?
In the ArcGIS Help you will find many mentions of raster layers. For example in the Applying pan-sharpening to a raster layer in ArcMap page.
Raster layers are layers but they are not feature layers, and there are many other types of layers that are not feature layers.
Even on the first page that you linked to it says (with my bolding):
For example, you can work with image layers using a special image classification toolbar.
These naming conventions are rooted in the world of object orientated coding, an object can inherit from a base object. So the base object is a layer and this will have properties that all other layer types will inherit, an obvious one is the name.
FeatureLayers and RasterLayers have many other data specific properties, but both of these inherit from Layer.
So Layer is not an abbreviation of FeatureLayer it is a base object (you could think of it as a simpler object) that other types of layers are inheriting from.