This is the approach I am currently taking. It is documented here: https://stackoverflow.com/questions/62774448/how-to-capture-cells-output-in-databricks-notebook

from IPython.utils.capture import CapturedIO   
capture = CapturedIO(sys.stdout, sys.stderr)
...
...
# at the end of desired output:
cmem = capture.stdout

I am writing the contents of cmem variable to a file in BLOB. BLOB is mounted to DBFS.

It would be good to see a working example supporting the @Hubert Dudek​ 's REST API approach that he mentioned above.