(...) and also the administrator of server is able to connect to the server locally there.

You have a big time difference between client and server. From the server's point of view the client is in the future.
This can cause several unwanted side effects like that a new created certificate is seen as not yet valid or getting strange timestamps.

When this test is done locally on the server machine, then it is obvious that client and server use the same system time and thus have no time discrepancy.

Fix the system time of the machine which is out of sync.
I hope this solves the source of your problems.