I'm not absolute sure how it works with a terminal server, but in general:
You have to start h2 in server mode and connect every client via jdbc to the server. So install the server on one machine and configure your client application to connect to the server. If the client is started twice every client connects remotely to the server via tcp/ip
http://www.h2database.com/html/tutorial.html#using_server
The mixed mode is a special variant of server mode and embedded mode. But if you start a client you don't want to start the server with the second client again.
So, try the server mode.