hi, when i change from 6 to 5, 5 to 4 or 4 to 3 either nothing happens for awhile as i keep pedaling and then it changes or i have to place additional pressure on the lever again, like is was changing gears again, and then it moves, i have read and watched the info you have but as i am such a newbie at this i want to be sure before i stuff it all up, everything else seems to work fine and i only had it serviced a month ago, it is a 2 tear old mtb and i am correctly using the right match of front and back cogs as you describe.
This problem has happened to me in the past and the way I fixed it was putting a few drops of some triflow in the shifter cable housings, then a couple on all the pivot bolts of the rear and front derailler, and cleaned the chain (this was just because it needed it). Work it while riding through all of the gears for maybe a block and the repeat the process, except cleaning the chain. If it still does then something will need replaced. Probably cables/housings and possibly the derailler itself. Which is caused from spring wear and break down.
Good maintenance to your Bike, can make it like the wheels are, true and smooth!