That's correct; automatic activation of unlimited turns occurs only at the time of purchase and only on the device used to make the purchase*. For the other devices, you must activate your unlimited turns manually. To do so, please follow the instructions on this page.
*The exception to this is Kindle Fire players using WhisperSync. If the state saved in the cloud included Unlimited Turns, the Unlimited Turns will appear when WhisperSync retrieves this state from the cloud. In other cases (e.g., if the state saved in the cloud does not include Unlimited Turns), you should do the manual activation.