No, hanetsugi is hane-connect. Often on 2nd line. The diagrams are not well aligned with the text. Hanekomi is wedging hane.
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A hane is a diagonal move from one of your stones that has 3 liberties (unless on 1st line) so that one lost liberty is an enemy stone and there is a cutting point.
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So 1 2 3 6 8.
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In terms of Go proverbs there’s
“attach, hane” which would the case of one opponents stone a lot of the time.
Then there’s “hane at the head of two and three stones”.
Also
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doesn’t this definition include number 5?
No, because it has 2 liberties.
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Japanese Hane? Chinese Hane? Common Usage Hane? 
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It does appear from this thread that the Japanese word Hane has a slightly wider use than the English word Hane, and there’s nothing wrong with that
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Well it is a Japanese word after all 
I was really just joking how for any topic we can debate ad-nauseam the variations of it.
Q: “What exactly is a hane”?
A: "An African hane or a European hane?
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What is the airspeed velocity of an unladen hane?
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Ohh I see, I misunderstood and was counting the liberties on the initial stone.
Now that I think about it, I guess I would only call 5 a hane if the stone to its right weren’t there
And I thought this would be the simplest topic of all 
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Is this from a book? What’s the source?
On the Online Go Semantics Forum?
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When do players talk about hane?
Mostly (if not only?) in this situation where black has to chose to go straight or put his stone ahead, at the contact of the white stone. And doing so leaving a potential cutting point.
On the left this is what I call hane (or figure if someone tells me).
Instead of going straight, black may consider other moves (like a one point jump.), it doesn’t really matter.what is important is the hane as being this local follow up.
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I think it is a type of atekomi (in Chinese we called it 擠, or 掖 if you really want to be specific, meaning hitting at the armpit, if this is a critical joint)
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“Dan magic”
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+1 atekomi.
Or “diagonal wedge”, as contrasted with the usual wedge in a 1-pt jump.
It’s a really cool shape when it happens.
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Dictionary of Go terms by Richard Bozulich, and the proverbs from Essential Go proverbs by John Power
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