Sure you can record from the output of the camera.
All those ways work but not the USB. The usb won't send the signal.
HDMI and SDI can output high definition. Component and composite only standard definition.
Regarding which recording device, there are a wide variety depending on form factor, small or large, battery or wall powered, multiple camera inputs or not. And whether it works with the software you are trying to use it with on your computer, for instance, just recording, or live switching, or web streaming.
I have used the atomos shogun as a standalone recorder, without the PC.
John is right, USB will not output live video, must use HDMI or analog. Slight correction, Component video WILL output 1080i if camera supports it, not limited to SD (Composite and S-Video are SD-only).
On a PC desktop, the BlackMagic Intensity Pro will work if you have an open PCI-e slot to put it in. Provides capture from HDMI or analog sources, plus output as well. Records as uncompressed or "Motion-JPEG" .avi - you will want to use the latter for sure, as uncompressed HD files require expensive drive arrays and are unnecessarily large.
Most cameras will send video direct from sensor to HDMI output before it ever gets compressed to camera format, so output is 4:2:2 uncompressed. Then you record to a 4:2:2 format with Intensity and really keep the quality up.
The Matrox MXO2 Mini is another capture device I rely on myself. Costs more, but provides a nice aluminum desktop breakout box, plus if you edit with Premiere it includes several plug-in effects.
EDIT: I'll second John's motion to look at Atomos, I use Ninja 2 and it works great to record HDMI source to ProRes 422, ready to edit.