  Microscopic mechanisms of the puzzling insulating ferromagnetism of
half-filled La$_{4}$Ba$_{2}$Cu$_{2}$O$_{10}$ are elucidated with
energy-resolved Wannier states. The dominant magnetic coupling, revealed
through evaluated parameters ($t$, $U$, and $J$), turns out to be the intersite
direct exchange, a currently ignored mechanism that overwhelms the
antiferromagnetic superexchange. By contrast, the isostructural
Nd$_{4}$Ba$_{2}$Cu$_{2}$O$_{10}$ develops the observed antiferromagnetic order
via its characteristics of a 1D chain. Surprisingly, the in-plane order of both
cases is \textit{not} controlled by coupling between nearest neighbors. An
intriguing pressure-induced ferromagnetic to antiferromagnetic transition is
predicted.
