Fill color: Why is there both Alpha channel and Opacity slider? When do I want to use one and when to use the other?

The SVG file format uses four different types of opacity in the style attribute:
- ‘fill-opacity’ specifies the opacity of the painting operation used to paint the interior the current object.
- ‘stroke-opacity’ specifies the opacity of the painting operation used to stroke the current object.
- ‘stop-opacity’ defines the opacity of a given gradient stop.
- 'opacity' specifies object/group opacity
<http://www.w3.org/TR/SVG11/masking.html#ObjectAndGroupOpacityProperties>

In Inkscape, these SVG style properties correspond to:
- 'fill-opacity': alpha slider (transparency) of the fill color
- ‘stroke-opacity’: alpha slider (transparency) of the stroke
- 'stop-opacity': alpha slider (transparency) in the gradient editor
- 'opacity': global (aka master) opacity slider (affects the object/group as a whole)

The difference between global opacity and transparencies of the fill/stroke attributes are most noticeable when applied to objects with a wider stroke: fill color and stroke color do overlap (stroke on top of fill color) which is visible when reducing the transparency e.g. of the stroke [1].
Global opacity on the other hand reduces the transparency of the object as a whole (or of groups).

[1] the filled area is defined by the geometric centerline of the object/path. The stroke is painted along this outline (centerline) of the object/path - half outside, half inside.