Gewehr98:
I guess this is a matter of terminology.
I've always refered to "take up" as the free movement of the trigger before the mechanism actually begins to engage.
For instance, the adjustments to an adjustable trigger are refered to as: "take up', "creep", and "over travel".
Take up is as above.
Creep is the amount of movement of the trigger after it actually engages.
Over travel is the movement after the trigger releases.
This is the terminology used in all my technical manuals and books, so it's the term I've always used.
I've worked on a fair number of Kahr's and I don't think your trigger should be "dangling back and forth loosely". Even though the trigger has a certain amount of free movement, it should be under reasonably stiff spring tension.
I have seen one Kahr with the very end of the trigger spring broken off that did this. The action worked fine, the trigger was just loose when at rest.
You might take a look and see if the spring tip is missing.
The older model Kahr's that lack the trigger stop surface on the rear of the trigger had a tendency to break trigger springs. These triggers can be pulled back until the trigger contacts the frame.
The newer triggers will stop well short of the frame, and this seems to have fixed the broken spring problem.