Fixed behavior of --search_data=quick and --search_data=complete

[schlunma]

Description

This PR fixes the behavior of --search_data=quick and --search_data=complete so it resembles the description in the docs.

I also fixed tiny typos and a mypy warning that showed when running pre-commit run --all.

Closes #2936

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[codecov]

Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 95.59%. Comparing base (d0eb4c3) to head (fc43775).
⚠️ Report is 4 commits behind head on main.

Additional details and impacted files

@@           Coverage Diff           @@
##             main    #2937   +/-   ##
=======================================
  Coverage   95.59%   95.59%           
=======================================
  Files         266      266           
  Lines       15573    15573           
=======================================
  Hits        14887    14887           
  Misses        686      686           

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