According to the FCC, a digital television signal is considered high definition if it outputs at 720p or higher. The FCC does not specify a minimum standard of quality for the final product, though, so it is still technically high definition television if a station simply rebroadcasts its low resolution analog content.
This is in important distinction because a similar problem exists in games. Some developers have chosen to reduce the framebuffer and effective resolution of their games in order to enjoy increased framerate, dipping into territory slightly below the 720p minimum standard. Then the framebuffer output is upscaled to a true HD signal, and as a result their product is still high definition. Going by the FCC standard, this is technically HD, but doesn't change the fact that the games run at sub-HD resolutions and are noticeably softer on high end televisions.
The term "High Definition Graphics" was first used as a marketing label by Sega to promote the increased resolution and 16-bit capabilities of the Sega Genesis / Mega Drive console.