test: Improve Error Reporting for Cluster Failures in End-to-End Tests

[systay]

Description

When running end-to-end tests, diagnosing failures during cluster startup can be challenging due to limited error details. This PR enhances error reporting by modifying how process information is captured and logged, allowing for clearer insights into failures.

These changes aim to reduce the difficulty in identifying the root cause of cluster startup issues, thereby improving debugging efficiency and test reliability.

Checklist

• "Backport to:" labels have been added if this change should be back-ported to release branches
• If this change is to be back-ported to previous releases, a justification is included in the PR description
• Tests were added or are not required
• Did the new or modified tests pass consistently locally and on CI?
• Documentation was added or is not required

[vitess-bot]

Review Checklist

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