There are two levels of iteration when an iterative solver is working on a nonlinear problem. To avoid confusion, let's call the iterative solver the PRECONDITIONED CONJUGATE GRADIENT SOLVER (PCG)
Both direct and PCG solvers can compute a nonlinear analysis. Nonlinear means that the stiffness matrix equation is solved once, then a convergence evaluation is made to decide if the nonlinear system has converged. If not another solution of the stiffness matrix equation is made with updated values. Each time the stiffness matrix is solved, that is called an iteration. After an iteration meets the convergence criteria, that is called a substep and the load is incremented and the next substep can begin with the first iteration.
As you note, the direct solver, also called the SPARSE SOLVER, uses LU decomposition to solve the stiffness matrix equation.
The PCG solver uses an iterative algorithm to solve the stiffness matrix equation. In a linear analysis, that is done once. In a nonlinear analysis, that is done for every dot on the N-R Convergence plot (every nonlinear iteration).