You need a cut-in box. They’ll have wings of some kind on them.

It looks like you already have an old work box, which is what you need if there is no stud. It's possible the wings just got pushed in and thats why it fell, or they ripped through the drywall because there is a chunk missing there. If you remove the fixture you should be able to see the screws to clamp it back and you can just rotate everything so the wings are secure behind undamaged drywall. (When you rotate the box you will have to adjust the fixture mount to keep it level)

This is correct

Now that I know what to look for, I can see the wings are snapped off. Must have given way. I assume I can just pick up another one at the hardware store, rotate so it’s behind some drywall, and pop it in?

Round ones are surprisingly hard to find. You may have good luck at Ace, but I had to order a pair from home depot a couple of weeks ago, which took a week to arrive.

My local Ace has them listed now in stock for ~$2 more.

Looks like our local Home Depot’s have a ton, thankfully.

That's good. Both my nearby HD and Lowes have been low on all junction box stock for a while now.

The "wings" are pretty generic. You could buy two single gang boxes (2 wings each) and transfer the 3 wings you need to your box.

And, it should be obvious from what happened, but don't drive the screws so hard that the wings break off or the drywall gets crushed! Just snug... Not the place to try out your new Christmas impact driver!

You will have a minor problem mounting your light fixture - some want the mounting holes to be in one location (vertical or horizontal) and some have rotating mounting plates that don't care how the box is oriented. If you have the first type, rotate the electrical box 180 degrees - since it has three wings, that should give them all some drywall to attach to. You might have to turn the power off and loosen the wires. If the light fixture is adaptable to any angle, then you have nothing to worry about...

Use some caulking around edges to help grab the box inplace with new box. Hopefully you get one that has the small lip. You could also use spray foam to help set it...but that can make a real mess of things if not careful.

As others have said the issue you might have is where your light fixture attaches to the electric box. If it has the round washer type flange that attaches to box and then screw from light connect to the washer flange, you should have play to rotate new box to get good grab. If the light attaches directly to box, you may have to get creative on repair.

Years ago I worked as an residential electrician's helper in high school and college. About 10 years in total.

If the light fixture is heavy you may want to place some thin metal/plastic stripping between the ears of the new Old Work Box and the sheetrock. Heavy fixtures tend to do exactly what this one did over time. Placing something rigid between the sheetrock and the ears prevents this by increasing surface area and keeping the edge of the ears from cutting through the sheetrock.

Most hardware stores sell some sort of flat strip with holes in it. Cut the strip to length, and use a finishing nail (very thin, easily pushed through/removed from sheetrock) to hold the strip in place while you place the box.

NOTE: Make sure the nail will not mar a wire.

TIP: Try to Find an area that will be covered by the fixture and place the nail there.

Once the box is installed and the ears are holding the strip in place, remove the nail.

There are other methods to keep the strip from falling into the void, but this one is easy if the holes can be hidden.

Simple. Toss the light and get a flush mount LED light bar. The mounting bracket should cover up that hole.