Simply because only ~300W/m^3 are being produced, that doesn't mean the reaction is not sustaining itself! Remember even though only ~300W/m^3 are produced per second, a cubic meter of matter in the center of the sun has an energy density of around 10^16 J/m^3. That's ignited plasma, no question about it.
And keep in mind, when you consider the 'Q' that various projects are envisioning, fusion reactors will be using most of their energy to maintain confinement (fed back into the lasers or magnetic fields); they are not going to be exploding like a thermonuclear weapon. They will have many times better yields than the P-P fusion chain, but the definition of ignition doesn't change. It's just the difference between burning incense or gasoline. At some point, either fuel "catches" and the oxidation becomes hot enough to maintain the kindling temperature for more oxidation to occur. Only difference is, it's nuclear chemistry instead of atomic chemistry.
This isn't right. A thermonuclear reaction is unstable, the reaction rate accelerates explosively as result of the heat it is generating.
The sun is like a furnace held together by gravity. Its stability is in stark contrast to a pair instability supernova, which consumes its fuel on a time scale of seconds.
I know in astronomy they refer to Hydrogen ignition and Helium ignition for the beginning of the different burning phases, but the ICF article puts "ignition" in quotes and carefully explains the positive feedback they are looking for. They have a link for "chain reaction" where the definition is given: "A chain reaction is a sequence of reactions where a reactive product or by-product causes additional reactions to take place. In a chain reaction, positive feedback leads to a self-amplifying chain of events."
Note: "self-amplifying", i.e. a runaway reaction.