chore: Removed deprecated LoggerProvider#addLogRecordProcessor()

[svetlanabrennan]

Which problem is this PR solving?

• Related to issue: [sdk-logs] remove deprecated LoggerProvider#addLogRecordProcessor() opentelemetry-js#5738

Short description of the changes

• Removed LoggerProvider#addLogRecordProcessor() in tests and readme and updated it to be set using the constructor option. See linked issue above for more details.

[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

Project coverage is 89.76%. Comparing base (b60b06e) to head (4c00019).
Report is 1 commits behind head on main.

Additional details and impacted files

@@           Coverage Diff           @@
##             main    #2895   +/-   ##
=======================================
  Coverage   89.76%   89.76%           
=======================================
  Files         187      187           
  Lines        9149     9149           
  Branches     1885     1885           
=======================================
  Hits         8213     8213           
  Misses        936      936           

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