I don't think it has anything to do with injection or Premium fuel. Beside DI there are also port injectors and if I am right, they are (indirect) operating when engine is not under heavy load. This is 99% of my driving. Engine itself if quiet and smooth if it is not engaged to driveshaft. For example first hundreds of meters after cold start, or idling in sport mode, it is very smooth. Almost the same as a 6 pot. But once in "gear" whole car shudders like espresso maker.It is more perceivable at low revs. Vibrations are often very irregular. Increasing revs to about 2000 and vibrations are gone. Above 3000 it begins to vibrate again. This sounds more like loos exhaust pipe. It doesn't bother me so much, because i almost do not use this rev range. After longer driving there is another type of vibration. Doesn't matter how many revs there are, whole drivetrain is vibrating like jackhammer. And then there is sometimes a bluster sound. Especially when revs suddenly drop to 1000 on a highway.
I am not a skilled constructor, but i have some points which can cause these problems.
1, Big engine, long stroke - physically must produce more vibration than same cylinder design of a half displacement.
2, heavy car, more load on engine - more vibrations
3, need for low emissions - too low revs - more vibrations. Older automatics didn't allow to go under 1300rpm.
4, need for elasticity - lighter flywheel - more vibrations
5, Atkinson cycle - May be? I think it rather reduces vibrations a little.
6, inline 4 cylinder configuration is the worst for vibrations. You can find many articles about that.
7, vibration dampers lost their capability. New car did almost not vibrate.
8, vibrations making all car parts worn. I Have problem with rattling or droning plastics around seat heating controls. The same and even worse can happen to important mechanical parts.
This evening on my way home, car was surprisingly smooth. Not the best, but better than yesterday.