I second the primer before skim coat.
The pic really makes this ceiling look bumpy.
It is a small ceiling area... it might be same time to take it down or rock over and mud. Your call,,, lots of high bumps might make this skim project very involved.
Lots of light is good for safety, but it can also blind you to the real condition of the surface you are working on. Check your work at the same light level it will be veiwed in. A hard lesson everyone must learn.
My lesson involved the underside of a warehouse roof.
I brought a 1000w Metal Halide flood light to light up the
painters work 35' up ( sissors lift and wand on the spray gun )
The paint job looked great until we turned the light off. LOL
PS. the trick to good mud work is less is more. Apply as little as you can.. 2 heavy coats does not = 3 thin coats.