Hand tightening: P
In this context, what does handshake mean?
It is the initial communication between two applications that will communicate. During the process of handshaking each of the applications will "negotiate" how they will work, one passes as you wish to communicate within certain specified parameters specified and the other says whether to accept or deny or offer alternatives to the other decide what you accept. If they can get agreement the communication itself starts.
The process can occur during data transmission after each step. One side can confirm receipt as agreed.
You have more details about HTTP in How does the HTTP protocol handle requests? .
In practice, what does "... your handshake interpreted by HTTP servers as an Upgrade request"?
This upgrade request is something that HTTP uses to indicate that it will use something new to the protocol, then HTTP is only used as transport.