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The sun is at ignition temperature, as defined by the ICF: Solar fusion is self-sustaining; that's all "ignition" means. It is fusion chain reactions which supply the heat necessary for the sun's core to be reactive.

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.

This isn't right. A thermonuclear reaction is unstable, the reaction rate accelerates explosively as result of the heat it is generating.

Sorry, but this isn't correct. There's no more "instability" in the ignition of a thermonuclear weapon than there is in the sun. The reaction kinetics are different because the reactants and confinement mechanisms are different. That's all. There is nothing fundamentally different about the physics which involves a release of binding energy from the strong nuclear force. The criticality, or runaway, reaction is entirely in the fission part of the bomb.

The "instability" in all the cases you're talking about is all about the containment's ability to confine fuel long enough to fuse; that's all. In a PPI supernova, most of the fuel is not instantaneously consumed by some sort of "instability." In fact, most of the fuel isn't consumed at all. The core is entirely consumed and blows most of the hydrogen/helium atmosphere (about 75% of the star) off into space. The same thing happens in a thermonuclear device. Ablation of the shell and x-ray pressure confines the reactants long enough for significant consumption of the reactants, but there isn't any special magic sauce there that isn't present in the sun. It's just a more energetic reaction because it isn't the P-P chain.