ci: fix playwright cache being ignored due to old version

[BobbieGoede]

🔗 Linked issue

❓ Type of change

• 📖 Documentation (updates to the documentation or readme)
• 🐞 Bug fix (a non-breaking change that fixes an issue)
• 👌 Enhancement (improving an existing functionality like performance)
• ✨ New feature (a non-breaking change that adds functionality)
• ⚠️ Breaking change (fix or feature that would cause existing functionality to change)

📚 Description

While trying to debug the test action I noticed we are using/used the ci.yml of vitejs as an example to set up playwright. After checking their current I found that they fixed an issue with the way the playwright browser is being cached which we still have (ref: vitejs/vite#11254).

Right now in the test job during the install playwright step, you will always see the old cached browsers being removed in favor of downloading the latest available version. This PR changes this to cache it and prevent redownloading, using the playwright version in package.json as cache key.

📝 Checklist

• I have linked an issue or discussion.
• I have updated the documentation accordingly.

[kazupon]

Thanks!