Custom label for code-block captions (Output 1.2 / Input 1.2 / Source 1.4 / Directory tree 1.2 / etc)

Is your feature request related to a problem? Please describe.
Although Listing 1.2 is used in English literature for both source code listing, pseudocode, programoutput, console sessions etc. I would often prefer to use "Output 1.2" when listing output, "Source 1.2" when listing source code etc.

I refer to the Listing in Listing 1.2 as the caption label. This might be the wrong terminology, I don't know. It is of course not to be confused with labels used with references.

Describe the solution you'd like
Input:

.. code-block::
  :caption: Output of nested loop.
  :caption-label: Output

  $ python -V
  Python 3.7.4

Output: Output 1.2: Output of nested loop.

Although not the prettiest, the code-block counters could be kept as they were, i.e. no matter the :caption-label: they would use the same counter producing what is in the screenshot.
I suppose the optimal thing would be to use separate counters for each :caption-label:.

Describe alternatives you've considered
I suppose one could use / create special extensions for each use case, extensions which would provide new directives like .. output-block::. This would be semantically pretty, but also a bit overkill in my view, after all it is just a label for the caption, it is not like the block itself should be rendered differently.

No files, tests, or entry points are identified in the issue. Start by locating Sphinx's code-block caption handling and counter logic, then determine how a custom caption label should interact with shared or separate counters; done means the requested directive option produces the demonstrated label and numbering.