It works!
I launched Pageant (PuTTY SSH authentication agent) and added the PuTTY private key id_rsa.pkk (exported from PuTTYgen (PuTTY key generator) from id_rsa ) to it. Now I can write my repositories.
A small guide for everyone who had the same problems:
0) Configure the local system to work with GitHub, as described in Creating SSH keys . How-to.
1) Open PuTTYgen , load / import the private key (OpenSSH) id_rsa and save it ( Save private key button) as id_rsa.pkk (in the same folder where id_rsa saved, ~/.ssh ). You need this PuTTY version of the key to authenticate PuTTY with Pageant and (if you use it) TortoiseGit.

2a) Open Pageant (on my computer it is C:\Program Files (x86)\PuTTY\pageant.exe ) and add the private key PuTTY. Or simply open the PuTTY private key file ( id_rsa.pkk ) with Pageant .
2b) For TrtoiseGit, I needed to install the SSH client ( C:\Program Files (x86)\Git\bin\ssh.exe ) in the global TrtoiseGit settings.

What is it.
It works for me, but I do not have a deep understanding of this thema and cannot provide a low explanation of the level of solution and the cause of this problem (and, for example, why it works for some clients and was not in some others). So feel free to edit / comment on this post if you can explain it all better.
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