"Geometric nonlinearity" is a confusing term, because it can include two separate effects.
In general you can write the global stiffness matrix of the structure as the sum of three terms: $K = K_e + K_\sigma + K_L$.
$K_e$ is the elastic stiffness, resulting from the stress-strain relationships in the material. For a linear elastic material it constant.
$K_\sigma$ is the stress stiffness which models the fact that the internal forces in the body (arising from the stress distribution) can do work when the body deforms. A simple example is a stretched string. If you pull a guitar string sideways to pluck it, the deformed shape is (approximately) two straight segments which are not collinear, each with the same tension as in the straight string. The resultant force of the two non-collinear tension forces opposes the sideways deflection. That is equivalent to a stiffness term which is proportional to the tension in the string.
$K_L$ is the "load stiffness" which is similar to the stress stiffness, but arises because the external loads on the structure may depend on the deformations. For example a pressure load acting on structure acts normal to the deformed shape of the surface, not to its original shape. Another example is a structure rotating at constant angular velocity about a fixed axis, where the centripetal force acting on each particle changes both direction and magnitude as the structure moves (the magnitude depends on the distance from the axis of rotation, and the direction always points towards the axis).
An example of $K_L$ is a model of a simple pendulum. The only stiffness term is $K_L$ caused by the fact the the external force (weight) acts at a different position when the structure moves.
A linear analysis with small strains and small displacements ignores $K_\sigma$ and $K_L$.
A "geometric nonlinear" analysis always includes $K_\sigma$ but usually ignores $K_L$. That corresponds to assuming the the strains are (infinitesimally) small, but the rigid body motion (specifically, the rigid body rotation) of the structure can be arbitrarily large.
I don't know any common term used for "including or ignoring $K_L$" In some situations (e.g the dynamics of rotating machinery) including it is essential. In other situations it is negligible and ignored.
A friction force between two surfaces is therefore a (nonlinear) load stiffness. If the surfaces move relative to each other, the friction forces do work and the direction of the friction force depends on the relative motion. But the magnitude of the friction force also depends on the stress component normal to the surface, and when the surfaces are not sliding there is no work done.
So it isn't very practical to include these terms in the global stiffness matrix somehow. In practice you would apply a nonlinear constraint to the model of the structure to create the correct friction forces in the model.