I think the key difference is that a camera lucida reflects the image directly into your eye, while a camera obscura projects the image onto a plane surface.
The simple CL in the above mentioned video is just a half-mirrored glass. It partly reflects your subject of interest, and partly transmits your drawing surface, so one seems to be superimposed on the other. The relationship of the two images changes if you move your eye, so you start by marking a few points of reference to help you hold the correct angle.
Because the camera lucida image is superimposed in your eye, there’s no direct way to record it, other than tracing. (BTW: I’ve encountered the viewpoint-shifting problem mentioned by BarnOwl myself. Not being an artist, I often make line art illustrations by tracing a photograph in Illustrator. If the subject is changed, I either have to take pains to shoot it from the exact same angle and distance, or I simply start over from scratch.)
A camera obscura uses a lens (or pinhole) to project light from the subject of interest directly onto a flat surface. The image is always the same as long as the subject, lens, and projection surface don’t move.
Cameras obscura (camerae obscurae?) are all around us, but we just call them “cameras.” The “hidden chamber” is the light-tight space behind the lens where the projection surface is placed. Usually the image is recorded on light-sensitive film or a digital sensor, but early cameras often had a ground glass at the film plane on which you could view the image directly before loading film. Before photo emulsion existed, you could trace the projected image by hand.
A very large camera might project the image on a wall, so you actually stand inside the camera to view the image, and the image is close to life size. There’s a portrayal of this in the movie “Addicted to Love,” in which the protagonist sets up a camera obscura in a room across from his estranged girlfriend’s apartment and watches her movements in the projection. The movie does neat things with the idea, although the effect might have been artificially enhanced. In reality I suspect the “camera” room would have to be very dark, and the girlfriend’s apartment would have to be very, very brightly lit, to produce such a large, bright projection.
Other projectors (for slides, movies, art, etc.), are basically the same thing turned around, so a small recorded image or drawing is projected as an enlargement. In fact, I gather some view camera models have been used to shoot negatives, then fitted with a light source behind the focal plane to make enlargements through the same lens.