Nothing profound, but hopefully useful to somebody?
I’m pretty sure the english term for “hane” is “turn”, though that’s also used to refer to the solidly connected one which makes a full triangle.
I have only heard it used when the stones are solidly connected. Are you sure that term can be used for a hane as well?
If turn can be used for both, is there another term for turn that can only be used for solidly connected stones?
Oops. I meant “bend”. “Turn” is used exclusively for the solid case, but “bend” is sometimes used for both.
Since this also is sometimes described as “bend around” or “turn around”, I agree with Bill Spight’s comment that it’s easier to use the well-established hane to avoid confusion.
Another distinction to note: a jump is typically used to refer to movement toward the center (perpendicular to the nearest edge), while extension is typically used to refer to movement along (parallel to) an edge. See also: https://senseis.xmp.net/?NamingExtensionsJumpsAndPincers
There are some terms from here that I’ve rarely or never heard used but are useful nevertheless, and could be worthwhile to know:
Bo-tsugi / pole connection: Connection of a one-space jump into a solid three-in-a-row
Fukurami / bulge: Making a tiger’s mouth over the opponent’s stones
Guzumi: A bad shape that is good in context
Hanedashi: A hane that invites a crosscut
Kikasare: Seems to mean a passive response
Maido: Seems to means a shape where a single third line stone is atari’d towards the edge from above
Souba: An even and reasonable sequence