Fix error messages and codes for timeouts

[agoujot]

Split messages depending on whether we or the server stopped the request.
Fix a helper function returning 500 in all cases.
Fix the timeout being stored where the status code should go.

Bug: T226307

[codecov]

Codecov Report

Attention: Patch coverage is 0% with 1 line in your changes missing coverage. Please review.

Project coverage is 71.04%. Comparing base (4b8eb28) to head (6539990).
Report is 94 commits behind head on main.

Files with missing lines	Patch %	Lines
src/EventSubscriber/ExceptionListener.php	0.00%	1 Missing ⚠️

Additional details and impacted files

@@             Coverage Diff              @@
##               main     #539      +/-   ##
============================================
- Coverage     71.30%   71.04%   -0.27%     
- Complexity     1275     1298      +23     
============================================
  Files            46       46              
  Lines          3792     3913     +121     
============================================
+ Hits           2704     2780      +76     
- Misses         1088     1133      +45     

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