Note: in this answer, I have taken "inaccurate" to mean inconsistent with whatever philosophy governs their behavior.
Well, going through them, we see that, when asked "Who are you?":
A will answer A, as he is an accurate truth teller.
B, in order to be inconsistent with the fact that he is a truth teller, can answer with anyone other than B, as he is inaccurate and cannot think he is B, but will tell you whom he truly (but inaccurately) thinks he is.
C is an accurate liar, and so when asked who he is, he will lie, and lie accurately, i.e, he can answer someone other than C. Thus, he can consistently answer that he is B.
D is an inaccurate liar, i.e., he will try to lie about who he is, but because he is inaccurate, in order to be inconsistent with his philosophy, he must name himself when he is asked who he is. If he were to name anyone other than D, then he would be an accurate liar. As the problem statement says, he is both deluded and dishonest, and so he will try to give you inaccurate information but can't - in other words, he tells the truth (without meaning to!)