Composite cables can't carry HD signals at all, don't see where you got that idea, even from the answers on that link. I believe composite is limited to 480i or less.
Component video can do 1080p, but most devices limit it to 1080i.
What kind of content are you trying to record from your computer? You may be able to use a screen capture program, or a different program to directly convert the movie file into a different type without copy protection. You may be better off looking around in the HTPC section here for help.