You’re almost certainly going to want gypsum plaster a.k.a. Plaster of Paris
Since you’re fixing a cracking issue, the last thing you want is to introduce something that won’t expand and contract like its neighbors. I’ve gotten away with this kind of repair using drywall mud in a pinch, but temperature or especially humidity swings make it less likely that modern mud compounds will move like the plaster around it and you’ll be back.
Remove any loose or oily material
I like to use a dish scrub brush or whisk broom. Dampen the surface before applying new plaster and watch for spots that don’t absorb uniformly. Remove the coating further on those.
If sections are found peeling away from the wallboard, carefully use a chisel to chip the edges back to solid.
That crack above the drapes will definitely need poked and pried to drop any pieces that have lost adhesion from the wallboard. Hit it with the wire brush, then if it doesn’t crumble back grab a putty knife and poke around. Odds are you’re going to have to make what looks like a bigger mess to actually fix the hidden damage.
The other crack may not be as bad, but it’s hard to tell from lighting. Same treatment.
Feather out like tape and texture
Unless your whole ceiling looks like a spiderweb (in which case congrats you’re probably in for removing and refinishing the whole thing), just feather it in. Be sure to pack all the voids. Sand smooth when done.
Good news is these are good candidates for localized repair. Poke around with your thumb to make sure things haven’t softened, remediate anything that has.
Long lines draw the eye so be extra careful. Sand the paint back a bit (is it lead?) around the areas you’ll be plastering. When you go to sand out the new plaster, use a longer sanding block than you think you can get away with and remember to use an X pattern.
Move the lamps around while putting on final coat and during sanding. Wipe up dust with a damp cloth before deciding spots are too high.
Are you sure you know why your ceiling cracked?
If it cracked due to old settling, great. If you have expansive soils, brace yourself. If you have changed the load characteristics of the building, it’s best if you can wait a few seasons to let it re-settle.
If you changed insulation or HVAC conditions, maybe also wait a few seasons. In-wall moisture changes are of particular interest since if you have gained a moisture problem you’re literally just hiding it by patching the cracks.
