Move tools cache directory

[AlekSi]

Description

See #4258.

Readiness checklist

• I added/updated unit tests (and they pass).
• I added/updated integration/compatibility tests (and they pass).
• I added/updated comments and checked rendering.
• I made spot refactorings.
• I updated user documentation.
• I ran task all, and it passed.
• I ensured that PR title is good enough for the changelog.
• (for maintainers only) I set Reviewers (@FerretDB/core), Milestone (Next), Labels, Project and project's Sprint fields.
• I marked all done items in this checklist.

[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

Project coverage is 74.10%. Comparing base (f95b5d2) to head (33913f6).

Additional details and impacted files

@@            Coverage Diff             @@
##             main    #4265      +/-   ##
==========================================
- Coverage   75.35%   74.10%   -1.25%     
==========================================
  Files         323      323              
  Lines       22407    22407              
==========================================
- Hits        16884    16605     -279     
- Misses       4303     4586     +283     
+ Partials     1220     1216       -4     

see 36 files with indirect coverage changes

Flag	Coverage Δ
filter-false	?
filter-true	67.15% <ø> (-1.55%)	⬇️
hana-1	?
integration	67.15% <ø> (-1.63%)	⬇️
mongodb-1	5.12% <ø> (ø)
mysql-1	?
mysql-2	?
mysql-3	?
postgresql-1	46.35% <ø> (-0.12%)	⬇️
postgresql-2	49.43% <ø> (-0.19%)	⬇️
postgresql-3	49.68% <ø> (-8.49%)	⬇️
sqlite-1	45.49% <ø> (-0.20%)	⬇️
sqlite-2	48.65% <ø> (-0.13%)	⬇️
sqlite-3	48.82% <ø> (-0.16%)	⬇️
unit	31.88% <ø> (ø)

Flags with carried forward coverage won't be shown. Click here to find out more.

[rumyantseva]