The phrases "offensive-minded" and "defensive-minded" are only relevant to the coach's background, but to say that's a single focus is overly simplistic. A good DC has to have a solid understanding of offensive strategies, since he needs to design schemes to defend against them, teach his players how to recognize certain formations, etc. Likewise, a good OC needs to know the nuances of defensive schemes so he can design plays to beat them, teach the QB how to read a defense, etc.
By the time a coordinator gets to be a HC, he's well-versed in offense and defense (and ST, for that matter), regardless of what his coordinator position was. In short, an effective tactician knows his opponents inside and out.
That's why the best computer security people are former (or current) hackers - they know the minds of their opponents.