I did some more experiments.
This time, I used a Qi charging set with a receiver that's said to output 1000mA.

The voltage while idle is pretty much exactly 5V.
I couldn't measure the voltage while loading due to the way I attached the receiver to the phone (see below).

The csd-tool reports charging currents up to ~780mA.
However, the current seems to fluctuate pretty much sometimes.

I am currently charging the phone via this setup.
So far, it charged for about 20% (from 50% to 70%) in about 30 minutes and about 40% (from 50% to 90%) in ~63 minutes.
This is about 1.43 times faster than with the touchstone kit.
The maximum charging current, however, is approximately 1.81 times bigger than with the touchstone kit.
The "slower than expected" charging could be due to the fluctuations in charging current.

This time, I didn't use additional cables to connect the receiver to the phone.
Instead, I used the flat cables that are already attached to the receiver.
However, the contact pins don't have the right polarity.
To correct this I cut the plastic between the flat wires and crossed the wires by bending one wire over the other.
Then I used a rasp/file to blanken the flat wires (remove the plastic cover) at the desired positions and used tin-solder to form new connection pads.
Finally, I could paste the receiver into the phone using the already pre-attached double-sided adhesive tape.
I added some more adhesive tape in order to push the connection pads down to improve the contact.

While charging the device gets pretty warm.
The location of the warm spot seems to be roughly underneath the camera.
I will try to keep an eye on that.

I attached some photos of all this.