Which electrode is anode and which cathode in a plasma reactor with high-voltage DC power supply?
Step back and look at the definitions of cathode and anode for a moment.

From the Wikipedia cathode page:


  A cathode is the electrode from which a conventional current leaves a polarized electrical device. 


From the Wikipedia anode page:


  An anode is an electrode through which the conventional current enters into a polarized electrical device. 


Conventional current flows from positive to negative.

In your diagram, the current flows from the high voltage positive terminal, through a spark gap, into the plasma reactor, through the plasma reactor, then out of the plasma reactor to ground - which completes the circuit.

That makes the place the current enters the plasma reactor the anode, and the other end is the cathode.  So, the pointed wire in the plasma reactor is the anode and the thick plate is the cathode.