Some airlines paint a spiral on their large "high-bypass" turbo fan engine cones. It is to warn birds and other critters (ramp personnel) that the engine is starting. Older jet engines are typically started by blowing air onto the turbine blades, so there isn't a whole lot of "noise warning" until it's too late. The intake "suction" on these high-bypass engines is not to be taken lightly. However, even at idle speeds, it's hard to note any "spinning motion".
And on these engines, the first set of blades are actually the first stage fan. They, and the cone, spin.

The MD-80 engine is a variant of the model that powered the original DC-9 and 727. They are turbo fans, but are not considered high-bypass. Their suction isn't as powerful. Being fuselage mounted, they're relatively higher off the ground, etc.
And on these engines, the first set of blades are called "stators". Neither they nor their attached cone will spin. It's hard to see because they're farther back in the engine nacelle, whereas a high-bypass fan is towards the front for all to see.
Looking at the video, the loose cone is bouncing against the stator "blades". If those were fan blades, the cone would've been shredded.

TL;DR Different engine types. The "cone" spins on some, but not all.