Hm, I've always taken that phrase to mean, "don't be prejudiced against a whole person: people are complex and you can probably find something to agree on with anyone."
So, I often try to say I don't like certain a thing about a person rather than saying I hate them. Hating everything about a person just feels a bit too simple and wrong. Perhaps that is too PC for these days, but, that's how I take it. Even Hitler had a mom- that sort of thing.
That said, I always seen the "don't hate the player, hate the game" saying the way GP does - i.e. shifting moral responsibility away from participants. IMO it does make sense in some cases, but does not in others. I see it as a spectrum of pressure put on individuals.
For instance, I wouldn't blame a customer service employee who lies to me because their boss ordered her to, and she will lose her job if she doesn't. For most of the population, losing a job is as close to life-threatening situation as you can get without an actual medical condition. OTOH, I will blame the boss who ordered lying to people. The boss has many more ways to choose from, and telling employees to lie to people shows preference for profitting by fucking other people over.