From what I understand the stone was supposed to resurrect someone (that's the reason it took it's name) but in reality it only brought back something like a ghost (as we see in the tale). I think this is confirmed when Harry holds the stone and his family seems to be alive BUT when he tries to touch them he can't. So, is it true? Could the stone really work but not in the way it's name describes?
Yes, yes it can. Not only is it possible for the stone to really work but not in the way its name describes, it actually--as you pointed out--does! On a more serious note, the stone is called that, I assume, because of the brother's wish. He told Death he wanted a stone to bring back people from death, and Death gave him this stone, which was subsequently named the Resurrection Stone. Also, Resurrection Stone sounds good, but I would not buy the Not-Quite-Resurrection Stone.