This all worked like a charm, until the latest Docker Desktop upgrade. Looks that the Docker now creates socket file in a user space. So, for the terminal, it was fixed by adding this line to the end of ./zshrc:
In PHPStorm, I had to define the same path in the "Build, Execution, Deployment :: Docker :: TCP Socket :: Engine API URL".
So, now I can use Sail in the terminal and in PHPStorm.
The remaining piece of the puzzle is the Tinkerwell - in the Docker tab I keep seeing "Cannot connect to the Docker daemon at unix:///var/run/docker.sock. Is the docker daemon running?" error and I don't know where to put my proper socket-file location exactly for the Tinkerwell to start using it…