Between's root word means "two." When you have more than two, you use among. Go back to shoes for a minute.
My friend's in a hurry. He flings open his shoe cupboard. He just grabs any pair from among all of them. He just selects from among all of them. Right? We both agree on that?
I'm in a hurry. I fling open my shoe cupboard. I don't care which of my two pairs of shoes I grab. I just pick from between them.
The OED examples seem forced. I wouldn't say "space lying among the three points." Nor would I write it. I'd be much more likely to write/say "space lying inside the three points." If I was striving to be as correct as possible, I'd say "the space lying inside the triangle defined by the three points."
Similarly, I'd say/write, "a treaty agreed to by three powers" or "a tripartite treaty."
As for the closed petals. How many petals are there? Seriously. Say it were some kind of a Venus Fly-Trap. The "petals" would number two. You'd put the needle BETWEEN the two petals. No one would say, "Take that needle, and put it among the two petals." No one.
Now, say it were a tulip and had five (I think tulips have five) petals. I don't know where the OED is getting its needles, but you can only put a needle BETWEEN two petals at a time. If you were to approach the tulip from above, could you, in that circumstance, say that you were placing the needle "among" the petals? I might agree to that one case. But I would probably backspace over it and retype "Place the needle so that it is surrounded by the petals."
Between describes conditions involving only two. So yes, with my limited shoe choices, I choose between my two pairs of shoes. My friend, not knowing I only have the two pairs, uses among.
(You know something. This is probably the most interesting thing I've discussed in about a month. Thank you very much for participating in this. God knows, I need the mental stimulation.)