Would it be better to use a higher diameter for the coil, and less turns ?The highest Q is achieved by using only 1 turn of very thick copper (magnetic loop antenna).Those antennas have a very high Q.
Or maybe a flat coil with only a few windings, like Tesla used ?

There are also commercial 13.56 MHz antennas available, for RFID applications, like this one: 

https://www.alibaba.com/product-detail/Free-Sample-Long-Range-PET-13_60597153218.html


I wonder why nobody detected any strange effects with this frequency, when it is so widely used in the industry.

Here's a thicker gauge variety of spiral:

https://www.aliexpress.com/i/32859268160.html

*hmmm*

It's not enough to illuminate a LED (only 250 mV_pp) but the strange thing is: the voltage does not depend on the distance between sending and receiving coil.   
(i varied the distance from 10 cm to approx. 1.5 m and in this range the voltage is constant)


I connected the sending coil to the generator (only one end, the other end open), generator was running at 13.44 MHz and 10V sinewave.
On the receiving coil i connected only one LED, no other diodes. What is the purpose of the normal diodes ? 

The LED itself is a diode so it should also work without additional diodes (?)

With a wire (30 cm) at the open end of the coil the receiving voltage increases to 350 mV_pp.
Ok, this is not a miracle, it's what ham radio operators do all the time: they are also using short vertical antennas with a coil at the base to tune it.
And everything i tested was near-field, less than one wavelength between transmitter and receiver.
It works even better when i change the frequency at the generator to 12.4 MHz.