[codex] Update Cake tooling

[AdamEssenmacher]

Summary

• Bump the local Cake tool manifest from 0.38.5 to 6.1.0.
• Replace deprecated Cake addins with local wrappers for CocoaPods, xcodebuild, lipo, and text/file helpers.
• Move build and pack aliases from DotNetCore* to modern DotNet* APIs.
• Update build docs to restore the checked-in tool manifest instead of recommending pre-1.0 Cake.

Validation

• dotnet tool restore
• dotnet tool run dotnet-cake -- --target=clean --dryrun --verbosity=quiet
• dotnet tool run dotnet-cake -- --target=nuget --names=Google.SignIn --dryrun --verbosity=quiet
• dotnet tool run dotnet-cake -- --target=nuget --names=Google.SignIn

The real Google.SignIn pack run produced 5 NuGet packages and completed externals, libs, and nuget successfully.