Yes.
They take the damage, because it's high as f%$~.
You can use the Grab An Edge Reaction (https://2e.aonprd.com/Actions.aspx?ID=95) to reduce the damage, though. It normally requires a free hand, but with Critical Successes, even characters with hands occupied (two weapons/shield, etc) can do it without much hassle.
Climb down 5 feet then drop.
There are also feats like the general feat Cat Fall. Or the Monk feat Dancing Leaf.
I would say you don't need to use a reaction, you can take an action to lower yourself over an edge, even if it's not directly stated as an option it makes sense.
It's probably a climb check to go over the edge and not immediately fall.
I agree with the answers re: standard PCs, but personally I've been wondering about PCs with special leaping abilities who can regularly fall from much higher w/o taking damage. Sure, they have to be able to fall after their special leaps for game balance/usefulness, but it seems odd they couldn't intentionally tap into that falling ability with a premeditated jump down.
"Didn't you just fall 30' w/o incident?"
"Yeah, but that was in the moment."
"You have those 'moments' most every battle."
"Spring theory?"
"Wuh?"
(Had a PF1 player whose high level Barb wore a Ring of Feather Falling simply because his frequent leaps hurt so much. That was quite a valuable slot.)
It is weird that you can sudden leap and take no damage, but that you try to make a similar jump without making an attack and would technically take damage.
However, RAW that is the case.
Ignoring that particular issue, you can climb over an edge and lower yourself down, reducing the height you fall.
You can also use any of the many things that reduce fall damage to help mitigate it, such as Cat Fall.
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The only time someone will be able to Leap high and then not take fall damage is Sudden Leap, otherwise they're still taking damage. You don't, as far as I can tell, actually jump that high when you jump far.
Flying Kick has similar rules regarding falling with no damage that Sudden Leap does.
But what Castilliano is likely referring to is things like Cloud Jump that let you regularly jump well over the limit of taking fall damage. Why wouldn't that also let you land safely? And if you can land safely when you jump that far from the ground, why not when you drop down from a high ledge?
Not trying to argue that it is not RAW to take damage after using Cloud Jump - just pointing out that it is really strange.
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I just have my players make a horizontal jump, applying the distance covered against the height of the fall. For example a jump result of 20 would knock 20 feet off of a 30 foot height.
Sure, walk off the edge... want to avoid taking damage and land on your feet?
A) Grab cat fall
B) use a reaction to grab the edge and then drop the remaining 5ft
C) use an action to climb (this can be dicy depending on the surface) and drop if you don't get a critical success
D) buy a bunch of cat talismans (unlikely that you will need it multiple times in a row)
Remember that 10ft is 3 meters.
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My own interpretation is that Leap can leap downward. It says, "If you Leap vertically, you can move up to 3 feet vertically and 5 feet horizontally onto an elevated surface." It does not specify up or down. And I claim that the character does not need to Leap onto a solid surface. Thus, a character who wants to jump downward 8 feet takes a Leap action to descend 3 feet and then they harmlessly fall only 5 feet.
Realistically, a downward Leap means that the character who jumped down prepared themselves for a hard landing using the same muscles involved in Leaping. It does not mean that they freeze in midair after a 3-foot descent and then fall from that standstill.
That only works for an 8-foot or shorter drop. High Jump can increase that to a 10-foot jump downward with a DC 30 Athletics check.
Climb and Grab an Edge handle longer descents for people without Catfall or similar feats.