Rear derailleur moves forward when pedalling stops

It looks to me like at the moment you stop the crank, the chain goes slack at the 'bottom' (normally the top) between the cassette and the chainring. The chain keeps on moving in the normal direction even though you stopped the crank. This effectively shortens the chain by pulling it through the derailleur, causing the derailleur to rapidly/violently take up the slack and jolt forwards.

The cause of this is likely some failure/jamming of the freewheel mechanism, where the rear cogs keep turning with the wheel but shouldn't (whether it's a freewheel or freehub design). Certainly that's where I would investigate, if there are no other apparent issues with the derailleur set up. If you can substitute in a different wheel then you might rule it in or out.

I'd say your freewheel is jamming but this other question is where the opposite happens and the cassette slips instead: Bike chain slips in a strange way

tl;dr it's the plastic disc between the spokes and cassette. Best fix is to just remove it.

As Swifty said, the problem is that the freewheel isn't running properly free, i.e. the cassette keeps turning with the wheel even though the chainwheels aren't pulling. This has happened to bikes of mine for two different reasons in the past:

• A really old, badly maintained (or, un-maintained) freewheel mechanism. Gunked up, grease spoiled, rusty. Not hard to see why this would seize up.
  Your bike looks in generally pretty ok condition though, so this is probably not the issue. (Even new freewheels can malfunction, but then it's more typically that they either don't engage properly or have constant too high friction, not this kind of seizing.)
• Many bikes come with a plastic disc between the wheel and cassette, often called “dork disc”. This disc is supposed to prevent the chain from dropping off the largest sprocket or the derailleur getting stuck between the spokes. That, if it does happen, is really messy; however, a properly set up derailleur should never make this happen in the first place.
  The problem (apart from æsthetics) with dork discs is that they're invariably made from brittle plastics, PMMA or something, and thus can easily get damaged in a way that gets the disc itself or splinters thereof stuck between the spokes and cassette. Indeed your dork disc is damaged, so I'm almost sure this is the problem with your bike. The result is exactly what you experienced: freewheel dragging along with the spokes instead of running free.
  Since the dork disc is not necessary as long as the derailleur works correctly (indeed good quality new bikes, at least MTBs, tend not to have one in the first place!), I would recommend you simply remove it. As it's broken anyway, the easiest way might be to clip it to small pieces and removing them with pliers, you may be able to do this without removing the cassette. A more proper way is to take off the cassette, then the disc, and put the bike together again.