refactor(common/logger): refactor and simplify logger storage logic

[XuehaiPan]

Description

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Motivation and Context

Why is this change required? What problem does it solve?
If it fixes an open issue, please link to the issue here.
You can use the syntax close #15213 if this solves the issue #15213

• I have raised an issue to propose this change (required for new features and bug fixes)

Refactor and simplify logger storage logic.

Types of changes

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• Bug fix (non-breaking change which fixes an issue)
• New feature (non-breaking change which adds core functionality)
• Breaking change (fix or feature that would cause existing functionality to change)
• Documentation (update in the documentation)

Checklist

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• I have read the CONTRIBUTION guide. (required)
• My change requires a change to the documentation.
• I have updated the tests accordingly. (required for a bug fix or a new feature)
• I have updated the documentation accordingly.
• I have reformatted the code using make format. (required)
• I have checked the code using make lint. (required)
• I have ensured make test pass. (required)

[codecov-commenter]

Codecov Report

Merging #216 (6e2ba34) into dev (d17aa77) will decrease coverage by 0.05%.
The diff coverage is 100.00%.

❗ Current head 6e2ba34 differs from pull request most recent head c430010. Consider uploading reports for the commit c430010 to get more accurate results

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@@            Coverage Diff             @@
##              dev     #216      +/-   ##
==========================================
- Coverage   97.20%   97.15%   -0.05%     
==========================================
  Files          98       98              
  Lines        4291     4293       +2     
==========================================
  Hits         4171     4171              
- Misses        120      122       +2     

Impacted Files	Coverage Δ
omnisafe/adapter/offpolicy_adapter.py	100.00% <ø> (ø)
omnisafe/algorithms/off_policy/ddpg_lag.py	100.00% <ø> (ø)
omnisafe/algorithms/off_policy/sac.py	100.00% <ø> (ø)
omnisafe/algorithms/off_policy/sac_lag.py	100.00% <ø> (ø)
omnisafe/algorithms/off_policy/td3.py	100.00% <ø> (ø)
omnisafe/algorithms/off_policy/td3_lag.py	100.00% <ø> (ø)
omnisafe/algorithms/on_policy/base/natural_pg.py	100.00% <ø> (ø)
omnisafe/algorithms/on_policy/base/trpo.py	90.14% <ø> (ø)
...mnisafe/algorithms/on_policy/first_order/focops.py	96.72% <ø> (ø)
...isafe/algorithms/on_policy/penalty_function/p3o.py	100.00% <ø> (ø)
... and 15 more

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[zmsn-2077]

After careful review, I agree with these changes.