tweak(github/ci): registry auth and lowercase image name

[tens0rfl0w]

Goal of this PR

Addresses two oversights from a previous change (#3212).

How is this PR achieving the goal

• The repository name and owner name are now converted to lowercase before pushing the builder image to the registry. This ensures compatibility with GHCR, which does not support uppercase characters in image names.
• Authentication with the container registry is now performed prior to pulling the image, addressing issues related to private access settings.

Note: While these changes have been tested, enterprise organizations may have different configurations when scoping access to private container images. As such, there is no guarantee that this will behave as intended across all environments.

This PR applies to the following area(s)

GitHub CI

Successfully tested on

Game builds: Not applicable

Platforms: Github-hosted Runner

Checklist

• Code compiles and has been tested successfully.
• Code explains itself well and/or is documented.
• My commit message explains what the changes do and what they are for.
• No extra compilation warnings are added by these changes.

Fixes issues

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