Simplify and deprecate commonerrors.WriteErrors

[AlekSi]

Description

Closes #2750.

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

Merging #3258 (7012ad6) into main (eccf40d) will increase coverage by 0.00%.
The diff coverage is 72.22%.

Additional details and impacted files

@@           Coverage Diff           @@
##             main    #3258   +/-   ##
=======================================
  Coverage   75.17%   75.18%           
=======================================
  Files         400      399    -1     
  Lines       22386    22378    -8     
=======================================
- Hits        16829    16824    -5     
+ Misses       4580     4575    -5     
- Partials      977      979    +2     

Files Changed	Coverage Δ
...ggregations/operators/accumulators/accumulators.go	84.78% <ø> (-0.33%)	⬇️
internal/handlers/common/update.go	89.39% <0.00%> (ø)
internal/handlers/commonerrors/error.go	88.23% <ø> (ø)
internal/clientconn/conn.go	55.84% <71.42%> (+0.05%)	⬆️
internal/handlers/commonerrors/write.go	84.21% <83.33%> (+3.61%)	⬆️
internal/handlers/commonerrors/command.go	94.44% <100.00%> (ø)

... and 3 files with indirect coverage changes

Flag	Coverage Δ
hana	?
integration	71.61% <72.22%> (+0.01%)	⬆️
mongodb	5.26% <0.00%> (+<0.01%)	⬆️
pg	64.66% <72.22%> (+<0.01%)	⬆️
shard-1	55.92% <61.11%> (+0.09%)	⬆️
shard-2	54.74% <66.66%> (+0.06%)	⬆️
shard-3	55.93% <72.22%> (+<0.01%)	⬆️
sqlite	47.00% <55.55%> (+<0.01%)	⬆️
unit	24.58% <16.66%> (+0.07%)	⬆️

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

[chilagrow]

There is a linter warning for deprecated function

[AlekSi]

There is a linter warning for deprecated function

Unfortunately, we can't fix it in this PR because of the way we use that linter and two separate configurations for golangci-lint. That code is considered new (it does not exist in the main branch), and our config disallows the usage of deprecated symbols in the new code.

That will not be a problem once this PR is merged.

[chilagrow]

Looks good to me 🚀