Don, did you ever figure out the difference between a "client" and a "server"?
And at the same time, a web browser is the client of computing and data servers in the cloud, provided over the network.
In reality, there can be many different client/server relationships existing in different directions over the same full duplex network connection. It's really a bi-directional multiplexed messaging channel, not a pure strict hierarchical relationship, and many client/server dependencies can go both ways over the same connection, at the same time. You can even run an ssh that opens tunnels in both directions, then tunnel X and http connections over that.
Once a connection is established, it really don't matter which end initiated the connection (or how many links and proxies and tunnels there are along the way) -- you're just sending messages both ways.
"Client/Server" is an over-simplified way of describing what's possible.
Within the context of the X Window System, the server is the body of code responsible for delivering the display to the user viewing it.
The server actually serves the user to the remote client.
It's like "you are not the customer, you are the product". The remote program needs to communicate graphically with you, and connects to the X server to do this; therefore, you are not the requester of a resource, you are the resource being served. :)
Cheers.