this apple TV business if very weird. I would have
thought HDMI is HDMI is HDMI, but clearly not.
HDMI comes in several revisions (up to 1.3 now) but the problem is more than likely related to HDCP rather than HDMI. HDCP (High Definition Content Protection) is a form of DRM that requires devices to communicate their purpose digitally to content providing devices. The intent is to prevent protected HD content from being passed to devices that can record it.
For instance, if your TV has a HD output & it is connected to a HD recorder, a source device digitally connected to the TV would poll it through the digital HD connection, discover it can record HD, & refuse to output protected digital HD content to it. (Optionally, SD content may be output.)
HDCP is proactive in that unless connected devices indicate they are
not recorders, or that they will not output HD content to them, digital HD output is not enabled. IOW, HDCP allows source-to-display-device protected HD transfers, but that's all.
This means that any problem anywhere in the device communication chain can shut down digital HD output, greatly complicating troubleshooting efforts. It is also why resetting or power cycling devices in different orders can mysteriously set things right.
Annoying, isn't it?