refactor: avoid ApiKeyOption in sample

[dbolduc]

Part of the work for #14759

Avoid using ApiKeyOption which we will remove shortly. We do not want to scrap the sample. Luckily we can achieve the same effect as ApiKeyOption using other options in our library.

Also, we should break out of the backoff loop when an attempt to authenticate is successful.

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This change is 

[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

Project coverage is 93.39%. Comparing base (a695357) to head (736c2da).
Report is 2 commits behind head on main.

Additional details and impacted files

@@            Coverage Diff             @@
##             main   #14760      +/-   ##
==========================================
- Coverage   93.39%   93.39%   -0.01%     
==========================================
  Files        2316     2316              
  Lines      207867   207870       +3     
==========================================
- Hits       194139   194136       -3     
- Misses      13728    13734       +6     

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[scotthart]

Reviewed 1 of 1 files at r1.
Reviewable status: all files reviewed (commit messages unreviewed), all discussions resolved (waiting on @dbolduc)