The flame one seems really weird to me in the English. A flame “starting/started” to go out only works in English if it’s a really big flame. Would 火が消えてきた be used for a match? And if so, wouldn’t the flame already be out at that point that phrase was used? Hmm…I guess if the match got fluttery there for a second that phrase feels right.

I like DBP’s explanation because it emphasizes that interval between states. And because something has been completed, some change has taken or will take place, maybe that’s why I feel like more qualifiers, either explicit or implicit, are required.

I guess basically I feel like “inceptive” and “reaching a state” are more closely linked than in the diagram she has at the end. It just feels like the “point of inception” is not as well-defined for the examples used in “reaching a state.”

The way to summarize it which covers all the bases most completely, I think, is to say that it signifies a process with emphasis on what has been or will be completed. Note that the process may or may not be completed YET. So, in the example sentence I suggested above, the speaker may continue to learn more about the Japanese mentality, but the emphasis is on what has already been completed. The same person might say, 「そのうち君もわかってくるよ」(Sono uchi kimi mo wakatte iku), which means, “Soon you will [start to/come to/etc] understand it, too.” This is future, but denotes that a process will gradually come to fruition.

Similarly, ～ていく (-te iku) is almost the same, but with emphasis on the beginning of the process. So, 「だんだん社会が変わっていく」 (Dandan shakai ga kawatte iku) means, “Society will gradually begin to change.” In certain cases this can be translated as “go on to” (like, 「あのときから親しくなっていった」(Ano toki kara shitashiku natte itta.) “After that, (someone) went on to become more intimate/closer.”), but this is a bit less directly equivalent to the Japanese phrase in my mind. The important thing is to understand that both emphasize a process, but they signify opposite ends of the process.

Because of this, incidentally, if you want to emphasize the process, and especially the completion of the process, ～てくる is useful. And the situation Daniel proposed with the mail is one such situation. So that’s a special use of this pattern, but it fits with the overall sense.