How do I set Apple TV HD to record a future program?

You would need to make sure that you are subscribed to their DVR service, which would be in addition to the TV package you have, you can verify within your account with them to see if that is setup. If you are and the app is not functioning as anticipated or you are unsure of what to do next then you will need to reach out to Spectrum.

The Apple TV itself is not what is used to record or store anything. It would be Spectrum’s DVR service that operates through the cloud, which would be initiated and accessed by using their app via the Apple TV hardware. You would need to download their app using the App Store on your Apple TV.

Also, It would still require you to subscribe to their DVR service, so if you only had a subscription to a TV package then you would still need the additional DVR add on in order to have that feature.

In the event that the app was not working correctly then they would need to issue an update.

Hopefully, you are able to get things sorted out. Best of luck

you can't, the storage is only for apps and cache while streaming

The only way would be if you used a live streaming service that had support for cloud DVR.