From the book “Real analysis: An introduction to the theory of real functions and integration” by Jewgeni H. Dshalalow (CRC Press, 2001, ISBN 1584880732, Chapter “10. Separation”, p. 182):
10.1 Definitions. Let $(X, \tau)$ be a topological space.
(i) $(X, \tau)$ is called a $T_0$ space if for each pair of points $x\not=y\in X$, there is a neighborhood of $x$, $U_x$ such that $y\in U^c_x$
(For a subset $A$, $A^c$ denotes the complement of $A$ (Ibid., p. 6).) Am I understanding it right that this definition is equivalent to axiom $T_1$ and therefore is incorrect?
You are correct. I recall the following definitions and explanations from [Eng].
References
[Eng] Ryszard Engelking, General Topology, 2nd ed., Heldermann, Berlin, 1989.