[PR #8991/1ba30112 backport][3.11] Update ClientSession reference

[patchback]

This is a backport of PR #8991 as merged into master (1ba3011).

Fixes #7515.

[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

Project coverage is 97.74%. Comparing base (ed5576d) to head (59f9505).
Report is 565 commits behind head on 3.11.

✅ All tests successful. No failed tests found.

Additional details and impacted files

@@           Coverage Diff           @@
##             3.11    #8995   +/-   ##
=======================================
  Coverage   97.74%   97.74%           
=======================================
  Files         108      108           
  Lines       34464    34464           
  Branches     4087     4087           
=======================================
  Hits        33687    33687           
  Misses        581      581           
  Partials      196      196           

Flag	Coverage Δ
CI-GHA	97.64% <ø> (ø)
OS-Linux	97.32% <ø> (+<0.01%)	⬆️
OS-Windows	94.89% <ø> (ø)
OS-macOS	97.01% <ø> (ø)
Py-3.10.11	97.10% <ø> (ø)
Py-3.10.14	97.03% <ø> (ø)
Py-3.11.9	97.26% <ø> (ø)
Py-3.12.5	97.36% <ø> (ø)
Py-3.9.13	97.00% <ø> (ø)
Py-3.9.19	96.94% <ø> (ø)
Py-pypy7.3.16	96.53% <ø> (-0.01%)	⬇️
VM-macos	97.01% <ø> (ø)
VM-ubuntu	97.32% <ø> (+<0.01%)	⬆️
VM-windows	94.89% <ø> (ø)

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