Fix compositional logging with lightning

[SkafteNicki]

What does this PR do?

Fixes #1756
The missing self._forward_cache was messing with the lightning logging.
Additionally, I have extended the integration testing quite a lot to be very specific what is logged when for future reference.

Before submitting

• Was this discussed/agreed via a Github issue? (no need for typos and docs improvements)
• Did you read the contributor guideline, Pull Request section?
• Did you make sure to update the docs?
• Did you write any new necessary tests?

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[codecov]

Codecov Report

Merging #1761 (f393232) into master (d396cda) will decrease coverage by 0%.
The diff coverage is 0%.

Additional details and impacted files

@@          Coverage Diff           @@
##           master   #1761   +/-   ##
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- Coverage      87%     87%   -0%     
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  Files         254     254           
  Lines       14306   14310    +4     
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  Hits        12513   12513           
- Misses       1793    1797    +4