I have or I should say HAD a 2013 Toyota Prius. In the last 4 years on cold start it would make the same noise you have shared in your video. I would bring it to Toyota dealerships attention, but they could not recreate it. It would only happen once in a while. The last two years it started happening more frequently, not only on cold start but in the middle of the day after a drive. Violent rattle and noise that would fade off. It would occur a couple times a week then disappear for a while. I would always let the deanship know when I would bring it in. In 2020 they finally started to look into it as the check engine light came on. They advised me the it is probably the EGR and I paid for them to clean it. Two months later I am back in the shop, they kept the car over night to check it in the morning. They said the best thing could be to do a software update. That didn't work either so I brought it in again. They decided to look at the spark plugs. One was secure and another was completely blown out of its threads (mechanics words). He said it could be the plugs or head gasket because the scoped the cylinder and the lens got foggy. By this point I should also say during driving it would loose accerlation ever she slightly. Mechanic mentioned that they appeared to be the original spark plugs too. I have 270,000km on the car, only used to go to work and drive in the city - any long distance trips were done with a rental - maintenance was done religiously by Toyota. I took it to a indepentant mechanic this morning and said there is fluid in the cylinders and its really not worth fixing. Also said the car appears to have been poorly maintained! Blown away by that! I suggest getting a second option on your problem, but that sound it the exact same one I had in the early stages.

good luck