[PR #8991/1ba30112 backport][3.10] 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 (7c3ff64) to head (eee89ed).
Report is 194 commits behind head on 3.10.

✅ All tests successful. No failed tests found.

Additional details and impacted files

@@           Coverage Diff           @@
##             3.10    #8994   +/-   ##
=======================================
  Coverage   97.74%   97.74%           
=======================================
  Files         108      108           
  Lines       34369    34369           
  Branches     4082     4082           
=======================================
  Hits        33593    33593           
  Misses        580      580           
  Partials      196      196           

Flag	Coverage Δ
CI-GHA	97.63% <ø> (ø)
OS-Linux	97.32% <ø> (+<0.01%)	⬆️
OS-Windows	94.88% <ø> (ø)
OS-macOS	97.00% <ø> (+<0.01%)	⬆️
Py-3.10.11	97.09% <ø> (ø)
Py-3.10.14	97.02% <ø> (ø)
Py-3.11.9	97.25% <ø> (ø)
Py-3.12.5	97.35% <ø> (ø)
Py-3.8.10	94.51% <ø> (ø)
Py-3.8.18	96.83% <ø> (ø)
Py-3.9.13	96.99% <ø> (ø)
Py-3.9.19	96.93% <ø> (ø)
Py-pypy7.3.16	96.52% <ø> (ø)
VM-macos	97.00% <ø> (+<0.01%)	⬆️
VM-ubuntu	97.32% <ø> (+<0.01%)	⬆️
VM-windows	94.88% <ø> (ø)

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