I guess most game players use Hauppauge 1212 HD-PVR because they like not only to record videos, but also to edit them and broadcast live gameplays. AFAIK you can't do it without a PC.

Hauppauge 1212 HD-PVR uses component video for HD videos only, but most users won't see any difference. And one can always use AverTV USB HD DVR, Blackmagic Intensity Shuttle or any other recording device supporting HDMI input.

The HDMI specs includes a copy-protection scheme known as High-Definition Copy Protection (HDCP). So to use HDMI (and get the HDMI certification logo on your device), manufacturers need to ensure that their products are HDCP-compliant. That pretty much means that HDMI inputs are limited to display devices (like TVs, projectors) and repeaters (AV receivers and switchers). By definition, that HDCP-encoded video signal is designed to be unrecordable. That's why there are few recorders with an HDMI input.

Component video doesn't have any protection, so you can use it to record HD videos.