I think this probably matches up reasonably well with what that book (認知と言語) were trying to get at. I think this also has some explanatory power when addressing why 言ってる・言ってた is used when reporting what someone else said. Like, “as far as I know based on what I saw/heard, they -said- that”.
I wonder what connection, if any, this interpretation has with the next question (which I’ll post in a second)…
No! It’s not true!
I address this a little bit after that description but I don’t expand on it so just to reiterate: Verbs can be categorised by lexical aspect. Some verbs actually straddle between different categories so are used in ways that may initially appear confusing or unexpected for a learner. This normally becomes evident when ている is used as many learners, especially beginners, strongly associate it with the English progressive form (-ing) which is misleading in many cases. 来る is one such verb. わかる is another.
As you mention, わかっている is normally used when someone is emphasising that they already understand something (“I got it already!”). Normally the lexical aspect of わかる is stative, so just saying わかる with no ている means that one is in the state of understanding (or the subject is in the state of being understood). However when わかっている is used it is acting as an instantaneous verb (so it is a binary of either having happened or having not happened) and the ている emphasises that the thing in question has already been understood and remains being understood.
Question Two:
What’s the difference between てある and ている?
Question two is inspired by this comment! Slightly less juicy this time so I won’t have to write a short essay again 
What exactly is the difference between てある and ている? Are they ever interchangeable? How do they relate to tense? Can you perhaps use the information about aspect from Question One to explain the difference? Does the answer change depending on the verb? Depending on the subject?
Here are two example sentences to act as a hint. You can also use them to try and explain the difference. Would you translate these sentences the same way? Why? Why not?
And a larger hint (and an answer to @I-k-d 's question) below:
Unfortunately 来てある isn’t grammatical. The reason being that てある is only used with transitive verbs, so an object is needed. That is, something needs to be plausibly markable by を, even if it is omitted. Also, the agent (the acting person/thing) needs to be something that has some will/can act of its own volition. And assuming the bus is meant to be the agent, normally a bus cannot do this…unless it is a magic cat bus… 
(Partial) source for this question: 日本語のルール
I am interested to see the thorough answer you always provide on this question as well. In general I think the ておく・てある・ている comparison is a big hurdle.
My understanding was that てある is describing the situation from the side of the object that has been actioned upon, but is saying it was put into that place by someone (presumably for a purpose) sort of like ておく except describing not from the doer’s perspective but from that of the thing acted on.
This would be why they can only be used with transitive or other move verbs, which ever you prefer to call them. As a self move verb wouldn’t make sense to be an action put in place by someone else, due to the nature of it being self move.
Sometimes I think this ends up being another case where Japanese sort of sounds like its in passive when you translate it to English.
ている on the other hand I think of as more of an objective statement about the current state of the verb/object.
So for
I would of say its like:
The watermelon is chilled/is cooling
The watermelon has been chilled
Not sure if this is really the right phrasing (chilled is weird in English since it implies its like frozen almost…but I’m not sure if we have a better word, “refrigerated” maybe?), but generally I would see it more as, at least for an English translation, the second sentence becomes more of a passive sentence the “has been” hints at the idea someone has done it, or put the process together so that it happens for one reason or another, the first sentence is more simply about the current state of the watermelon itself, not the action/state that has been put upon it.
I am somewhat confident, but definitely think I can still have details wrong, I am going to avoid your larger hint spoiler for now and revisit once more people have answered. Looking forward to it, questions like these are great to see what everyone thinks and their impressions on the grammar points.
I’m gonna post an answer and my comments on Saturday. In the meantime, if you aren’t sure what the answer is or why exactly the answer is what it is then please also post questions!
Also, why use てある instead of ている or ておく? What particles can be used with てある? Does it change the nuance at all?
EDIT: I have not forgotten! Just been busy with work. Will get get back to this when I get time (probably the weekend, hopefully before).