Design A takes the basic core and adds a piston driven by a simple clock. The repeaters should be set to at least 14 ticks to avoid fruitless thrashing by the piston. (The cobblestone needs time to form). The design is demonstrated in this video.
Design B sends its cobblestone upwards. Rather than a clock, the piston is triggered by actually sensing when the cobblestone is created. The piston is sticky, and holds onto a "switch block" which must be pushable but transparent — that is, can push the cobblestone, but not carry redstone current. Top slabs, etc., will work as well as glass. It is demonstrated in this video.
The second repeater must be set to at least delay 2, or the piston will drop its switch block. If this happens, the generator can be restarted with the button. If the generator fills its pillar, it will stop until the newest cobblestone is cleared. Besides mining it directly, this can be done by mining any block(s) in the pillar and pressing the button. The pale stone brick indicates a few optional blocks below the fluids.
- The water and lava can be covered with slabs, especially once cobblestone forms.
- The redstone dust directly behind the piston isn't actually needed for the normal cycle, but it is needed for the button to be where it is. You can alternatively put the button in there instead of the dust.