This version of solid fuel lighter takes a different approach. The fuel is binary, having separate fuel and oxidiser components - in order to be extinguishable.
The fuel, likely a metal of some sort - probbaly a magnesium aluminium alloy or similar, and the oxidiser, likely a solid nitrate of some
sort - both come in extruded rod form. The cross section of the rods interlinks, so the oxidiser and fuel rods can be indexed against each other, and they interlink to provide maximal interface area between fuel and oxidiser. Indexing will be via lever or wheel on the side of the ligher. The idea being it burns at one end of the rods, and the rods can be indexed against each other, nominally moving the fuel as it's consumed, and moving the oxidiser to control burn rate.
Igniting it might be a bit tricky, but once going it should burn quite nicely, especially if you aim the oxidiser decomposition temperature somewhat below the flame temperatue (but not too low...). When you want to put it out, withdraw the oxidiser back into the lighter so that the "working face" telexcopes down away from the burning face of the metal/fuel rod. The fuel material has to be flamable enough to burn in the presence of the decomoposing oxidiser, but should rapidly fizzle out once the oxidiser is withdrawn. Obviously some experimentation is needed to get this right.
Think of it as a thermic lance, but with a solid rod of oxidiser rather than piped in O2.