ci: Fix multiple issues: refs, scripts, variables, locations

[joeyparrish]

This fixes multiple CI issues:

• Since moving to pull_request_target trigger (ci: Change test trigger #37), we were checking out the wrong ref in build.yaml.
• Repository variables should be used instead of secrets. This new feature of GitHub was introduced in January 2023, and gives us an easy way to configure the repo without abusing the secrets system. https://github.blog/changelog/2023-01-10-github-actions-support-for-configuration-variables-in-workflows/
• Move the api-client folder to the root where it is more discoverable.
• Move the build matrix config file to the root where it is more discoverable.
• Move package installation steps out of the workflow and into a script where they can be tested in a PR.

[joeyparrish]

Unfortunately, this isn't triggering the test itself because the new trigger (pull_request_target) runs the workflow that exists on the target (shaka-project), not the fork (joeyparrish). So the effect of this isn't seen until after it is merged.

However, you can see the effect here on a dummy PR on my fork:

https://github.com/joeyparrish/static-ffmpeg-binaries/actions/runs/6341990006/job/17226831542?pr=5

Configure Build Matrix logs:

{
  enableDebug: true,
  enableSelfHosted: false,
  matrix: [
    {
      os: 'ubuntu-latest',
      os_name: 'linux',
      target_arch: 'x64',
      exe_ext: ''
    },
    {
      os: 'macos-latest',
      os_name: 'osx',
      target_arch: 'x64',
      exe_ext: ''
    },
    {
      os: 'windows-latest',
      os_name: 'win',
      target_arch: 'x64',
      exe_ext: '.exe'
    }
  ]
}

You can see the effect of vars.ENABLE_DEBUG.

After this is merged, we should see the effect of vars.ENABLE_SELF_HOSTED on the repo in shaka-project, after which I can do another manually-triggered build and verify that everything works on linux-arm64.

Above I mentioned that pull_request_target runs the workflow that exists on the target (shaka-project), not the fork (joeyparrish). So one good side-effect of splitting the build scripts out from the workflow into separate script files (#22) is that this enables pull_request_target to test changes to those scripts.

[joeyparrish]

I discovered some further issues that impact PR testing, so I've fixed those here and rewritten the PR description to match. PTAL!