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[Akoto]

What's that brown thing below the wagon? Tree trunk? ASCII supremacist here.

Yep, that's a tree trunk in Mayday.

[PatrikLundell]

As mentioned, you can adjust the width manually, but obviously starting with a water course with the desired width means no adjustment is needed.
However, that's not related to the depth of the fall. Is the level of the lower part of the river level with the ground further downstream? If that's the case it's an issue of the river dropping abruptly while the ground drops gradually. You could try having the river at the higher elevation for longer, although that may result in an aqueduct instead:

33R33
33R33
00R00
00r00
00r00

or
33R33
00R00
00r00
00r00
00r00

Still assuming a 3*3 embark, but I don't think you have any problem with extending the logic to 5*5.

[Akoto]

Is the level of the lower part of the river level with the ground further downstream?

No, it's a straight drop from the picture I posted above down to the bottom of the waterfall, at which point the river continues onward.

[PatrikLundell]

If the lower part of the river is all in a canyon I suspect the river elevation is lower than the "normal" elevation.

Edit:
I tried 4 different variant on completely flat terrain (river flowing from south to north), and they all were as expected:
- Aqueduct: A wacky case where a river flows in the air and then drops 3 Z levels (with trees growing under the river...) down to the level of the normal terrain. This is essentially the bottom one in my previous post.
- 3 tile baseline: the first one posted. Terrain and river raised 3 Z levels on the last embark line and the line just south of that. Again, the water dropped 3 levels to the plain below.
- 1 tile baseline: Only the tiles of the elevated river had the ground elevated as well (providing some additional plains level terrain to each side. Water dropped 3 levels here as well, although there was a little spill to the sides (but shouldn't be enough to cause flooding, as it should evaporate).
- 1 Z level: The previous one, but just a single level elevation difference. Didn't seem to produce mist, though, but no water spill either.

In all the cases the river was adjusted to go from x = 20 to X = 29 through all the river embark tiles and the tiles to the north and south of the embark to ensure a straight course.
Note that in all the non aquifer cases you'd have to do some digging on at least one side to reach the water fall from the plain as there's a little canyon effect as the waterfall drops abruptly, while the terrain drops gradually.