Crisp up the wording around error handling clients

[benjie]

Description

Following on from #10323, this helps by better defining what an "error handling client" is (since this is new terminology) and gives examples at different layers of granularity. It further clarifies additional text that might be interpreted ambiguously or confusingly.

Type of change

Documentation only.

How Has This Been Tested?

Documentation only.

Checklist:

• I have followed the CONTRIBUTING doc and the style guidelines of this project
• I have performed a self-review of my own code
• I have commented my code, particularly in hard-to-understand areas
• I have made corresponding changes to the documentation
• My changes generate no new warnings
• I have added tests that prove my fix is effective or that my feature works
• New and existing unit tests pass locally with my changes
• Any dependent changes have been merged and published in downstream modules

Further comments

cc @eddeee888 and @martinbonnin

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[eddeee888]

Thank you! This helps expand my understanding of "error handling client"