feat(#673, #523, #848): back-port authjs migration

[phoenix-ru]

🔗 Linked issue

Relates to #673, #523, #848

This is a (partial) back-port of #818 to current code. Because completely updating to @auth/core makes the module very unstable, we decided to instead back-port the advancements made in the PR to also prepare for a future update.

Differences with #818 :

• no @auth/core version bump;
• internal helpers are preserved because they seem to work better than these of @auth/core;
• no breaking changes.

❓ Type of change

• [ ] 📖 Documentation (updates to the documentation, readme or JSdoc annotations)
• [ ] 🐞 Bug fix (a non-breaking change that fixes an issue)
• [x] 👌 Enhancement (improving an existing functionality like performance)
• [ ] ✨ New feature (a non-breaking change that adds functionality)
• [x] 🧹 Chore (updates to the build process or auxiliary tools and libraries)
• [ ] ⚠️ Breaking change (fix or feature that would cause existing functionality to change)

📚 Description

📝 Checklist

• [x] I have linked an issue or discussion.
• [ ] I have added tests (if possible).
• [ ] I have updated the documentation accordingly.