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High-quality single crystals of Ni₃(MoO₄)₂TeO₃ were synthesized via chemical vapor transport and verified using single-crystal X-ray diffraction (SXRD). Magnetization data revealed strong anisotropic magnetism, with two antiferromagnetic transitions at T_N1 (~17 K) and T_N2 (~14.5 K) along the H//b direction. Notably, T_N1 exhibited characteristics resembling ferromagnetism in the H⊥b direction. Neutron measurements disclosed a net magnetization along the b-axis for the Ni zig-zag chain, while adjacent chains along the a- and c-axes exhibited antiferromagnetic coupling with magnetic propagation vector of k = (1 0 0.5). The Ni(I) magnetic moment was aligned with the b-axis, whereas the Ni(II) site exhibited canting with components along ma, mb, and mc, contributing to the complex magnetic behavior of Ni₃(MoO₄)₂TeO_3.  A sharp magnetic-induced spin-flip transition occurred at H_C (critical magnetic field) of 2.5 T for H//b orientation. Furthermore, combined specific heat and magnetization measurements confirmed an unusual low-temperature shift of T_N2 with an external field up to H_C in the H//b direction and T_N2 transition disappeared for H > H_C.