Nope. In X, your monitor is the server. In your example, the laptop is the X server, and the desktop is the X client. HOWEVER, and probably what trips people

Nope. In X, your monitor is the server. In your example, the laptop is the X server, and the desktop is the X client. HOWEVER, and probably what trips people up, is that the desktop is your COMPUTE server. But the laptop is the PIXEL SERVER. I mean, sure, you can call your laptop the client all day long, but in X windows terminology the laptop is the X server. So you'd be wrong. :-)