*But I am not even sure if this is correct. And the details eludes me. For example: How do I discover what name to include in myservice.services.yml?*

Now that Drupal 9 is end of life you no longer need to do that. Replace the argument list with the autowire option:

*mymodule.services.yml*

    services:
      mymodule.my_service:
        class: Drupal\mymodule\MyService
        autowire: true

and in the service class:

      /**
       * The date formatter.
       *
       * @var \Drupal\Core\Datetime\DateFormatterInterface
       */
      protected $dateFormatter;
    
      /**
       * Constructs \Drupal\mymodule\MyService
       *
       * @param \Drupal\Core\Datetime\DateFormatterInterface $date_formatter
       *   The date formatter.
       */
      public function __construct(DateFormatterInterface $date_formatter) {
        $this->dateFormatter = $date_formatter;
      }

So to discover the service you only need the interface DateFormatterInterface. Use autocomplete and the IDE will automatically add a use statement at the top:

    use Drupal\Core\Datetime\DateFormatterInterface;

See the change record: [Core services provide aliases for autowiring][1]


  [1]: https://www.drupal.org/node/3323122