I would suggest going with a bar loading screen which basically informs the user that content is loading and you are preparing them for an unique experience. You can also try to engage the user by informing them about the game and the experience offered
For example Empire total war informs the user about what he can experience in the game
This game Nyrthos gives historical information about the game background
This loading screen provides information about the characters the user can play in the game
This screen provides tips on how to be successful in the game
That said, I also recommend looking at this article which talks about Final Fantasy XIII shows a recap of the plot so that you know what you are getting into which informs users about the story and also helps them remember if they havent played the game for a while
Finally, after enduring many years of “hey can someone explain the ending of this game to me” with Final Fantasy games, Final Fantasy XIII has made a simple change that has made me really happy. As soon as you start loading the game, recent plot events are relayed to you on the loading screen! Let’s have a quick look at this small but much appreciated usability-enhancing feature.
This really makes me happy. The game is taking a loading screen – otherwise a waste of the user’s time – and puts on it a couple of valuable paragraphs recapping the game’s plot. For me it also has the benefit of being able to see some of these names in writing. When the game uses words that are both odd and very similar – “l’Cie”, “fal’Cie”, “Pulse fal’Cie”, it helps me keep things straight to see them spelled out.
I strongly recommend against providing an alternate game in the loading screen as you are distracting the user from the focus of the game and are not utilizing the time he spends waiting in enhancing his experience and enthusiasm about the game (as the above examples have done). Also apparently Namco Bandai holds the patent to have minigames on their loading screens only so unless you work for them, you are opening yourself to legal issues.
Edit : Based upon your update, since this is a simple game, I would still recommend a simple bar screen but provide information about how the user can play the game (like shortcuts or codes and so on). For example taking this web cricket game, in the loading screen it just basically shows me the controls
You can also use the screen to provide tips as shown below