Passwordless access to raspberry pi via ssh - ssh

Question

I can't setup passwordless access to Raspberry Pi. I am using similar instructions like this one. But it still asks for password when I
ssh to my raspberry. I made this with other servers and everything worked ok, but not with my raspberry (both of them).
Thanks to Jakuje, become a little bit clearer:
successful debug log:
debug1: Offering RSA public key: /home/tmp/.ssh/id_rsa
debug3: send_pubkey_test
debug3: send packet: type 50
debug2: we sent a publickey packet, wait for reply
debug3: receive packet: type 60
debug1: Server accepts key: pkalg ssh-rsa blen 535
debug2: input_userauth_pk_ok: fp SHA256:XPQA6tgcRCam/L18YP4ccPwsOsGtO65ippi7kua4mTM
debug3: sign_and_send_pubkey: RSA SHA256:XPQA6tgcRCam/L18YP4ccPwsOsGtO65ippi7kua4mTM
debug3: send packet: type 50
debug3: receive packet: type 52
debug1: Authentication succeeded (publickey).
fail log:
debug1: Offering RSA public key: /home/tmp/.ssh/id_rsa
debug3: send_pubkey_test
debug3: send packet: type 50
debug2: we sent a publickey packet, wait for reply
debug3: receive packet: type 51
debug1: Authentications that can continue: publickey,password
debug1: Trying private key: /home/tmp/.ssh/id_dsa
debug3: no such identity: /home/tmp/.ssh/id_dsa: No such file or directory
debug1: Trying private key: /home/tmp/.ssh/id_ecdsa
debug3: no such identity: /home/tmp/.ssh/id_ecdsa: No such file or directory
debug1: Trying private key: /home/tmp/.ssh/id_ed25519
debug3: no such identity: /home/tmp/.ssh/id_ed25519: No such file or directory
debug2: we did not send a packet, disable method
debug3: authmethod_lookup password
debug3: remaining preferred: ,password
debug3: authmethod_is_enabled password
debug1: Next authentication method: password
Why it ignores id_rsa key and asks for id_dsa ???

Answer 1

debug1: Offering RSA public key: /home/tmp/.ssh/id_rsa
debug3: send_pubkey_test
debug3: send packet: type 50
debug2: we sent a publickey packet, wait for reply
debug3: receive packet: type 51
debug1: Authentications that can continue: publickey,password
says that the key was tried but not accepted by server. There will be a problem in server configuration/permissions. Running the server in debug mode identifies where the problem is. The owners and permissions on home directory, .ssh directory should be the following:
chmod g-w /home/your_user
chmod 700 /home/your_user/.ssh
chmod 600 /home/your_user/.ssh/authorized_keys