We report on a comprehensive characterization of the newly synthesized Cu2+-based molecular magnet [Cu(pz)2(2-HOpy)2](PF6)2 (CuPOF), where pz=C4H4N2 and 2-HOpy=C5H4NHO. From a comparison of theoretical modeling to results of bulk magnetometry, specific heat, μ+SR, ESR, and NMR spectroscopy, this material is determined as an excellent realization of the two dimensional square-lattice S=12 antiferromagnetic Heisenberg model with a moderate intraplane nearest-neighbor exchange coupling of J/kB=6.80(5) K, and an extremely small interlayer interaction of about 1 mK. At zero field, the bulk magnetometry reveals a temperature-driven crossover of spin correlations from isotropic to XY type, caused by the presence of a weak intrinsic easy-plane anisotropy. A transition to long-range order, driven by the low-temperature XY anisotropy under the influence of the interlayer coupling, occurs at TN=1.38(2) K, as revealed by μ+SR. In applied magnetic fields, our H1-NMR data reveal a strong increase of the magnetic anisotropy, manifested by a pronounced enhancement of the transition temperature to commensurate long-range order at TN=2.8 K and 7 T.