Spin part of the angular momentum from the Lagrangian

The division of the total angular momentum into orbital and spin parts is unphysical, a bit like how the division of energy into potential energy and kinetic energy is unphysical. (Are static electric and magnetic fields purely potential energy while electromagnetic waves are purely kinetic? This is not a physically meaningful question, so we don't ask it.)

Only the total angular momentum is a physically meaningful quantity, and to define it in terms of the stress-energy tensor is to put the cart before the horse. The proper way to define the total angular momentum is by applying Noether's theorem to the group of rotations. In the case of a Dirac field, one indeed obtains the expression for $mathcal{J}_z$ given by the OP.

The Belinfante–Rosenfeld correction term mentioned by the OP is defined in terms of the spin tensor which comes from Noether's theorem, so to define angular momentum in terms of the stress-energy tensor would be a circular definition. However, provided that this correction term is in fact included in the definition of the stress-energy tensor, the OP's expression will yield a sensible total angular momentum density tensor and will thus "include spin", whatever that means.