Thank you for the great advice! I feel really grateful to those who've spent their time not only watching but analysing, too. Here's a longer video of a friendly match with my doubles partner, recorded this month. I'm the guerrilla in black t-shirt.
Apart from me being slow and bald, I also noticed my high serve was actually pretty low. Will surely work on improving that. As always, any tips and feedback from you guys will be much appreciated. My main focus now is on lowering the stance, adding the split-step without thinking about it and of course losing weight. Badminton is very complex and sometimes I feel lost knowing how many aspects of my game I need to improve but not being aware of what to prioritize.
I would also agree to work on your movement. Try to time your split step at the same time as when the person on the other side of the net will hit the shuttle. Try to bend your knees so that your stance is lower on the split step (pretend that it is a squat). Then regular four corner movement patterns with lunges to the front of the court. If you can get to the shuttle more easily, it will be a good foundation to hit better shots.