I attached a picture with my confusion. Hopefully, someone will be able to shed some light upon this.
When you claim f(u[n] - u[n-4]) = f(u[n]) - f(u[n-4]) you are claiming that f() is a linear function. Is it? Why are you saying that f(u[n-4]) = (n-4)u[n-4] You have f(x[n]) = n * x[n] Whatever you plug in as the argument of f(), that only replaces the parameter x[n] in the expression.
Just a quick comment that the step f(u[n-4])=(n-1) u[n-4] is the incorrect step as mentioned above. The step f(u[n] - u[n-4]) = f(u[n]) - f(u[n-4]) is correct because this is a linear time varying system.
I figured it out. I was confused between the actual model and the input-output relationship that arises from it. Thanks for the answers.