I tend to go with your second interpretation there. Miller has shown through his work on Sin City that he has a very specific idea about what the proper responses to "crime" and "social degeneration" are, and they tend to be summed up by this kind of mindless, vengeful rage. Batman in DKR seems to be embodying this kind of "anyone who commits crime has chosen to be hurtful towards others, so anything we do to them is fair game" brutality, which is directly at odds with how I feel about the Batman character (in the manner of one thinking of a close friend, as Ms. Fall describes above).
Of course, that's just Dark Knight Returns. Dark Knight Strikes Again, where Batman murders Dick (after telling him "you were never good enough") among other acts of dubious heroism, seems to be an entirely different matter. At that point Miller didn't seem to be interpreting the DC heroes so much as just pissing on them. Which I also didn't appreciate, but I solved that problem by putting down DKSA after I read it through the first time and never, ever picking it up again.