What "all focusing points" means is that each and every focusing point is associated with the metering zone(s) that includes the area(s) which that AF point covers. Some of the areas of sensitivity for each of the AF points straddle more than one metering zone. In that case each of the metering zones included in a particular AF point's area of sensitivity are affected by whether that AF point is 'in-focus' or not.
It doesn't mean that all AF points are given equal weight for every time you meter, it just means that every one of the AF points is linked to the specific metering zones that cover the same area as that AF point. At any given time, if a particular AF point registers as 'in-focus', then the metering area linked to that AF point will receive weight based on that.
Does evaluative metering put more weight on the active focus point?
Sort of. It puts more weight on the AF points that are confirmed to be 'in focus.' So it's not whether they are active or not, it's whether they are actually in focus or not.
If you are using a focus mode with more than one focus point selected, the camera will focus on the area of greatest contrast it detects within any of the areas those active focus points cover (Hint: the area of sensitivity for each AF point is larger than the little rectangle you see in the viewfinder).
The metering area(s) that contain the focus point that detects that area of greatest contrast will get the most weight in 'Evaluative' metering. The metering areas linked to any of the other active focus points that are also confirmed as 'in-focus' will also carry more weight than other 'non-focused' AF points. The metering areas linked to AF points that are active but not confirmed as 'in focus' at the time will not get any special weighting due to their status as active focus points.
What about multiple focus points, e.g. when using the automatic selection of focus points SEL [ ] which might yield something like in the picture below? Is there more emphasis put on those metering zones that "map to" the active focus points?
See above. There's only one area of greatest contrast that the AF system is locked in on. If any other AF points are lit up in the viewfinder it means that they are also confirmed to also be sufficiently in focus to be considered 'in-focus'. But only one area of contrast within one AF point is actually being used to focus the lens. The camera searches for that area of highest contrast from among the coverage areas for all of the active AF points.
Apart from autofocus information, there are a myriad of other factors that determine the weighting of each metering zone when using Evaluative Metering. As the EOS system has developed since it was introduced in 1987, Evaluative Metering has also evolved. It's gotten better with age as processing power and memory capacity have increased tremendously, not to mention the advent of color sensitive RGB+IR light meters.