  We report on a comprehensive characterization of the newly synthesized
Cu$^{2+}$-based molecular magnet [Cu(pz)$_2$(2-HOpy)$_2$](PF$_6$)$_2$ (CuPOF),
where pz = C$_4$H$_4$N$_2$ and 2-HOpy = C$_5$H$_4$NHO. From a comparison of
theoretical modeling to results of bulk magnetometry, specific heat, $\mu^+$SR,
ESR, and NMR spectroscopy, this material is determined as an excellent
realization of the 2D square-lattice $S=1/2$ antiferromagnetic Heisenberg model
with a moderate intraplane nearest-neighbor exchange coupling of
$J/k_\mathrm{B} = 6.80(5)$ 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 $T_\mathrm{N} = 1.38(2)$ K, as revealed by
$\mu^+$SR. In applied magnetic fields, our $^1$H-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 $T_\mathrm{N}
=2.8$ K and 7 T.
