Is it illegal if you create a backup of an HDD that has copyrighted software installed on it?
This could be anything from legally purchased games and or expensive popular graphics/video editing programs, just as an example.
Is this backup an illegal copy of the original HDD?
You're allowed to make backups of copyrighted software, as long as you are authorized to use the software, the backups are not distributed, and they are destroyed when/if you are no longer authorized to use the software.
(a) Making of Additional Copy or Adaptation by Owner of Copy. -- Notwithstanding the provisions of section 106, it is not an infringement for the owner of a copy of a computer program to make or authorize the making of another copy or adaptation of that computer program provided:
(1) ...(not very helpful)...or
(2) that such new copy or adaptation is for archival purposes only and that all archival copies are destroyed in the event that continued possession of the computer program should cease to be rightful.
The law is usually used to justify copies of the installation media, but it would be unconscionable to cut your rights off there and make backups illegal for machines containing copyrighted software. After all, with such an interpretation, no Windows machine could be backed up since Windows is copyrighted.
Copyright law (17 USC 117) specifically allows this:
it is not an infringement for the owner of a copy of a computer program to make or authorize the making of another copy or adaptation of that computer program
if you are the owner of a copy of the computer program, provided:
(2) that such new copy or adaptation is for archival purposes only and that all archival copies are destroyed in the event that continued possession of the computer program should cease to be rightful.
There seems to be a bit of a meme out there that you can make only one backup copy (here, here, here) but the plain language of the statute does not support that interpretation, and "all archival copies are destroyed in the event..." would be incomprehensible if you were only allowed to make a single backup.
Only software enjoys this additional permission to archive. The distinction between "program" and "data" is not entirely clear, but the definition of computer program ...
a set of statements or instructions to be used directly or indirectly in a computer in order to bring about a certain result
can be reasonably interpreted to include datafiles that are read in order to make sounds or display images. (See the US Copyright office FAQ regarding non-program materials and backups).
