Chef resource to register gitlab runner automatically
Your second link just use the official cookbook. This official cookbook still uses the LWRP syntax, so you need to check in the provider directory for the resource action code -- which for register is basically just two execute commands (extract from the file linked above):

converge_by("Register runner #{new_resource.description}") do
  execute "#{env} gitlab-runner register --non-interactive #{options}"
  # giving time to gitlab-ci to manage the request
  execute "sleep #{new_resource.sleep}"
end


So mostly, your exemple ends up calling the gitlab-runner register command with the options defined in the options hash (with a little transformation of the hash into command parameters).

There's no 'core' chef resource to register gitlab runners out of using an execute resource to run the command.

A basic example of this resource is present in the cookbook Readme:

gitlab_ci_runner 'my runner' do
  options({
    registration_token: '1234567890',
    url: 'http://gitlab-ci.myinstance'
  })
end