On all my jobs I've gone first, and have came back to do just the baseboards last. My painting order is a bit unorthodox to most, in that I always do baseboards last at the end of the job totally, as I find I get a cleaner cut by painting the baseboards to the wall vs cutting in at the wall to the baseboard. It also saves time when doing the walls as you don't need a 100% perfect cut at the wall. Using the light and shadow trick, you can be more off with white to the wall vs wall paint on the baseboards (the baseboard's own shadow will cover 1-2mm of white, along with the caulking/etc and the baseboard usually having big gaps to the wall anyway) and of course gravity itself ends up pulling your cuts down if you're not mega ultra careful cutting your baseboards in with the amount of paint on the brush. It also is good on a long job to have a relatively easy day at the end and finalize the job and tie up loose ends on said easy day of baseboards + putting everything back together.
Having done some vinyl and ceramic tile installs (not for "pro" customers, just my own house and friend's houses) I personally think baseboards should be installed last if they're a normal 3" or so baseboard. You can't really use shoe molding on those, since they're already too small. It's better to install them at the end of the flooring process and get them flat on the new flooring, vs shoe molding or worst, just leaving them the way they are and having ugly huge gaps under them. In ceramic's case, it prevents the tiler from getting grout on the baseboards like a doofus or grouting to the baseboard (cracked tiles and grout possible with that) and looking ugly, too. If it's an older/more upscale house with bigger 6-8" baseboards, then you can use a shoe molding, but shoe molding on small baseboards looks really bad to me. I'd also rather be the one pulling the baseboards, too, to be able to hopefully not rip the walls up too badly, and feather in the bottom so there's not caulk lines/ripped drywall/etc. If I have to be one installing them it's whatever, too, because inevitably I'd have to go after the carpenter and "fix" it all anyway. As a (very) amateur trim carpenter I tend to use the GRK Torx head trim screws and a drill, no nail gun.
I've had a few dings/etc to fix from floor people but never had touchup issues with such tiny dings (1/2" or less.) Worst flooring doofus spilled a whole energy drink on a freshly painted wall, though. :/ And a tiler got grout into freshly painted baseboards and textured a couple of inches of them. :/