This is very subtle, which explains why nobody else has answered. It has to do with the logical connection between noun and verb, which is made "narrower" due to the syntax of these phrases. The "trick" is that your sentences are too short, that is, they are incomplete, not being appropriate for written or spoken English, making the differences between the (a) and (b) verbs seem less apparent. Nobody says, ""There ruled a King" by itself, it does not make sense. It does not have sufficient information. If we rewrite the sentences to be correct, then (more or less) the differences can be sensed; the (a) verbs seem to be more logically connected and "expected", and "complete", compared to the (b) verbs, which seem "incomplete". Let me rewrite them to be proper English, the way it would be actually used: 1) "Between the two rivers, in lands otherwise unknown, there ruled a King." [correct] or "Between the two rivers, in lands otherwise unknown, there lived a King." [correct] or "Between the two rivers, in lands otherwise unknown, there reigned a King." [correct] 2) "Between the two rivers, in lands otherwise unknown, there complained a King." [incorrect]. or "Between the two rivers, in lands otherwise unknown, there loved a King." [incorrect]. or "Between the two rivers, in lands otherwise unknown, there killed a King." [incorrect]. Why? The (a) verbs strike me as more "expected", logically connected, and complete the sentence correctly. The (b) verbs are confusing. For one thing, "complained" is not a kingly verb; Kings are not known for complaining, so the verb does not match the noun. This makes it more striking to the reader and harder to understand; if we said, "the king complained" it would be easy to understand. But the syntax of the sentences prevents that, making things more complex. Connected to this reason, is another, ambiguity; "there killed a King" is confusing; it could mean the King killed someone: "there killed a King his advisor".

This is difficult to explain, because it's not something native speakers think about at all, but I'll try. For the king question, "there" is moved to the beginning of the sentence to emphasize the place that the king ruled. You could just as naturally say "A king ruled there," but then the emphasis is on the king rather than on the kingdom. It's not natural to say "there complained a king" unless there is some reason you need to emphasize the place where the king was complaining. Such as, "In the smoking ruins of the bombed-out castle there complained a king." Even that sounds kind of silly, and it wouldn't be natural to say it that way. With a verb like "complain" it's more usual to emphasize who is complaining, so you would put "king" at the beginning of the sentence. For the ship example, if it appears, it's there. If it disappears, it isn't there. So you really can't say "there disappeared" -- it can't be there and not there at the same time. In the jewels example, "steal" and "snatch" have more or less the same meaning when used as verbs, but "stolen" is a special adjective. Once something has been stolen, you can refer to it as stolen. "Police recovered the stolen items," or "That man was driving a stolen car." We don't use "snatched" in the same way. I hope this helps.

Thank you for reading and answering my questions, Christine, Sebastian, and Robert! You helped me and gave very long comments. I chose Sebastian's answer for the best answer because he gave me detailed comments. Thank you. I will study to write my graduate thesis harder.

I don't think I have much to add to Sebastian's very through analysis, but I would add one thing. Definitely in the case of (1) and (2) and to some extent (3) (depending on context,) these constructions would not typically be used in either spoken language or casual writing. They sound like something that would appear in very formal writing, or in a narrative. (1) sounds like it came out of a fairy tale or fantasy story and (2) might appear in a narrative as "And suddenly, there appeared a ship on the horizon." Perhaps the latter could appear in spoken language if you are telling a story in a compelling or poetical way. This is perhaps why the construction is so restricted, as Sebastian described. Constructions that are not familiar to us from everyday speech and writing will be hard to process unless they are part of a set phrase like "Neither a borrower nor a lender be" or the meaning is clear from context, as Sebastian described. Unfortunately, there are often no set rules or if there are, well I guess you will have to figure them out for your thesis! English especially has been beaten up over time and there are a lot of weird little things left over. Let me know if you have any more questions! I find this stuff fascinating.

(3) a. "On the table next to the bed, there were stolen jewels." [correct] b. *""On the table next to the bed, there were snatched jewels." [somewhat permissible but not good syntax] c. "On the table next to the bed, there were jewels that had been snatched from the store." [correct] d. "On the table next to the bed, there were snatched jewels from several department stores he had visited (at nighttime) recently." [correct] Here, the same principals apply, though it is not easy to narrow them down exactly. "Stolen jewels" can be used because it is apparently to the reader of what the meaning is; those are jewels that have been stolen. Essentially, stolen is more passive of an modifier than snatched, and more obviously connected--everyone thinks of "stolen jewels", since thieves like to steal small, highly valuable items. Whereas, "snatched" adds a sort of anticipation and ambiguity, making the reader confused--snatched from where? See my "3d" example sentence for a solution that preserves the original, but takes away the anticipation and ambiguity, while leaving some gramatically correct surprise. Maybe a professional will come along and reduce my explanation to a simple sentence; otherwise, this is a bit of a game, all languages evolve in this manner, with a subjective logic that makes perfect sense in context (of a native speaker) but confusing to an outsider.