Refactor GitHub Actions workflow for publishing to PyPI and TestPyPI

[LacombeLouis]

Description

Checkout issue #793.

Type of change

Please remove options that are irrelevant.

• Breaking change (fix or feature that would cause existing functionality to not work as expected)

Checklist

• Make sure to update the pypi and testpypi Trusted Publishers in the settings of your PyPI and TestPyPI accounts.
• Configure GitHub Environments named pypi and testpypi in your repository settings.
• For pypi the environment should require manual approval (recommended by PyPI Trusted Publishing).
• After switching to Trusted Publishing you should remove any long-lived PYPI API token secrets from the repo and revoke them on PyPI/TestPyPI as described in the PyPI guide.

Guidelines

• I have read the contributing guidelines
• I have read and followed the testing guidelines

Quality Checks

• Linting passes successfully: make lint
• Typing passes successfully: make type-check
• Unit tests pass successfully: make tests
• Coverage is 100%: make coverage
• When updating documentation: doc builds successfully and without warnings: make doc
• When updating documentation: code examples in doc run successfully: make doctest

Contribution Documentation

• I have updated the HISTORY.rst and AUTHORS.rst files

LLM Usage (Optional)

If you used a Large Language Model (LLM) to assist with this contribution, please provide details:

• LLM used: (e.g., Mistral, Claude, GPT, Gemini, etc.)
• Purpose: (e.g., code generation, documentation, test writing, refactoring, etc.)