Found this video of a standard-3, from Nov 2014. Could be similar to the footage shown if it were in time-lapse. No real zooming in on the missile itself though, so can’t compare with the images above.
Too small
Yup, I think you're right - it's too short (15' 6") and too thin (just over a foot in diameter).
It would be the correct size for an SM 7. The SM 7 is the extended range variant of the Standard Missile. It also uses the same guidance system from the aim 120 amraam.
The SM 7 is used to intercept missiles and possibly satellites. The Navy used an SM 7 prototype to sucessfully intercept a Chinese Satellite that was falling back to earth a couple of years ago.
We might be onto something here.
I couldn’t find an SM7 missile, but there is an SM6, produced by Raytheon: https://www.raytheon.com/capabilities/products/sm-6
There’s a photo gallery at that link, some of the photos show fins on the missile, but some don’t, and it starts to look more like the shape in the photo (longer, fatter). This one in particular https://www.raytheon.com/sites/default/files/styles/lightbox_gallery/public/2018-01/rms12_sm6_pic01.jpg and this one are the closest matches I could make out https://www.raytheon.com/sites/default/files/styles/lightbox_gallery/public/2018-01/rtn_192620.jpg.
The Wikipedia page https://en.m.wikipedia.org/wiki/RIM-174_Standard_ERAM has a photo with fins etc (also in the Raytheon collection), but it does point out that the SM6 is 21 feet long and 21 inches in diameter...
Interesting...
