Ceiling – this texture called and how to replicate it

Priming then painting: The texture is water based, so you need to prime first with oil or shellac based primer. This will seal the texture and "waterproof" it.

Use plenty of ventilation and possibly an organic respirator.

Once primed, any paint can be used (make sure the full drying interval is used).

Removing: will work before any painting. Put down waterproof tarp, wall-to-wall and taped 6" up. Use a misting hand sprayer or a garden sprayer. Let soak 15 min, respraying if dry spots develop. Scrape with a long handled pole and a stiff angled putty knife. Keep the knife flat and start slowly until you get a rhythm. No sense in making another repair for gouges. Clean any residue with a new sponge mop and some TSP or a light detergent.

That is in fact a "sand finish", your intuition was right.

Now depending on the age of the house it could be sand added to the plaster, added to the mud, or a "sand finish" paint. The last is your best hope of matching, as you would drywall it all smooth then simply use brushstrokes to recreate the pattern. There are premixed sand paints amazingly enough.

When I've seen this done with wallboard mud, they have a mixer on site to stir every so often (the sand settles slower than blueberries). A quick google search reveals a number of pros complaining about how hard sand finishes are to match. Practice practice practice, on some scrap drywall.

Using a piece you cut off the ceiling for repair, try to figure out what layer the sand was added at (plaster, top coating on drywall, paint).