Mechanics behind auto-ignition temperature
To be clear, I'm no expert on the ASTM standards writers' thinking, and all I have to go on is the standard you've posted.


Yes, that's exactly right.  Some of the reaction's heat will be lost to the surroundings, but enough of it will go into the reaction mixture to raise its temperature and promote further, more rapid, combustion.
There are reactions that produce lots of heat but no flame, hydrogen peroxide decompostion, for example.  The standards writers had to come up with a clear definition for ignition, and as you note, they did: "the appearance of a flame accompanied by a sharp rise in the temperature of the gas mixture."  They even say that ignition is a subjective term, so they are careful to define it clearly.  There might be some chemical reaction that produces a large amount of heat in air with no visible flame (even in the dark, as the standard requires) and it might even qualify as an ignition event to you or I or any other sensible person watching, but it would not fit the standard.  They have to set some limit, after all, and that was what they set.
My original thinking on this was flawed; I misunderstood the air temperature reference.  The reason that they mention the air at all is that they are being specific that the reaction mixture should be an air/fuel mixture, rather than something else (oxygen/fuel, for instance.)  The predetermined temperature aspect of that point has to do with the AIT determination procedure itself.  They are setting the temperature of the system to some value T, then adding the fuel, then waiting for ignition.  So yes, the air in the flask is more or less in thermal equilibrium with the substance being tested.  It can, however, leak out and exchange with the air outside the flask. If the substance does not autoignite, the flask is purged and the temperature increased (including the air in the flask.)  I'm not sure whether they want a completely new sample of the substance for each temperature measurement.  It seems likely, since we're dealing with small amounts of generally volatile compounds.