If the server is started on the same terminal, then it's the server's stderr that is presumably being written to the terminal and which you are not capturing.
The best way to capture everything would be to run:
script output.txt
before starting up either the server or the client. This will launch a new shell with all terminal output redirected out output.txt as well as the terminal. Then start the server from within that new shell, and then the client. Everything that you see on the screen (both your input and the output of everything writing to the terminal from within that shell) will be written to the file.
When you are done, type "exit" to exit the shell run by the script
command.