It appears that a star is used, while in space, with the laser being used as an artificial star. I only see this directly mentioned in the video (1:24), and indirectly in the text, "thus acting like a source of artificial starlight". I guess this makes sense, as it's the only way to do it "end-to-end" while in space.
There are actually two stages of calibration in space. The first stage is aligning the mirrors, using direct measurement of the mirror alignment with the interferometer, after they are unfolded from their unaligned stowed positions.
> To measure the shape of the Webb telescope’s primary mirror, engineers use a test device called an interferometer, which shines a laser down onto the mirror.
The next phase you’re referring to also uses a laser, but this time it’s shone off the mirrors into the actual scientific observation sensors, hence they describe it as being used as a source of artificial starlight.
On the ground, you can easily place a laser in an optically identical position as a star (near infinity). In space, I don't think this would be possible, since it would require that the laser, and the optics that would be required, would all be within a fraction of a wavelength of the true optical path. In the end, you would still need to align to a star anyways.