Two things, you didn't show the actual pushing in of the collets to release the filter pipes, these can be real pigs, and it would be a good idea to press the water dispense lever or button on the fridge to release any residual pressure.
For some reason your video omitted the important step of pushing the collar around the water pipe where it enters the filter IN while at the same time pulling on the water pipe in order to release it.
The next time just connect it up run 3 litres of water through the dispenser your just purging the pipes /filter of air that’s why your still getting splashing at the dispenser after you’ve run the water through the filter . There wouldn’t be any nasty material coming through that filter the way your doing it as it’s a filter that’s it’s job it’s just air your getting rid of .
great walkthrough. also your point is really good about let the water flow for some minute before connect the new filter to the fridge. if you don't do that the particles and the coals inside the new filter may go to your fridge solenoid valve and clog it.