Gaussianity of lattice models in statistical mechanics

Recently there was a result on triviality (or Gaussianity) of the Ising model and $phi^4$-theory in dimension $d=4$. This therefore holds in any dimension $d geq 4$.
We also know that the 2D Ising model is not trivial.

However, at the same time the 2D Ising model can be mapped to the (transverse field) 1D Quantum Ising model which again can be mapped to free fermions using the Jordan-Wigner transformation. The free fermions are free and therefore Gaussian.

So these two notions of Gaussianity are obviously not the same. What can resolve my confusion?