So, I just want to add some more feedback.

First of all, the placement may be an issue.
I experimented a bit with the Qi setup that I showed in my last post and this setup also seems to be sensitive with respect to the placement.
I observed the charging current with the csd tool and it seemed that the current depends on how the phone was placed on the sender pad.
However, I am not sure if my observations are truly exact and if I really observed the true cause-effect relation.

The touchstone setup is also pretty sensitive with respect to the placement.
In this setup the placement problem, however, is to a large degree solved by the magnetic locking mechanism that pulls the receiver in place.
So, take care how you place the four small metal discs around the receiver when experimenting with the touchstone.
From my experience, it's best to just use the original "plastic foil" from inside the cover as this features four "dents", one for each of the discs.

Another observation I just made:
I gave the touchstone another try and used a more powerful power supply this time (1.5 A instead of 1.0 A).
Now, I could observe charging currents up to 780 mA with the touchstone.
An interesting point, though, is that the charging current seems to decrease over time.
At first, I could observe about 700 mA which went down to 600 mA and now reached about 450 mA.

Another observation, the thermal issue I observed with the Qi setup seems not to show up with the touchstone variant.