Shouldn't linear time-invariant systems be called affine time-invariant systems instead?

A system is some kind of function that maps an input as a function of t to an output. $$y(t) = H(u(t))$$ This system is linear if the following holds: $$y_1 = H(u_1), quad y_2 = H(u_2)$$ $$alpha y_1 + beta y_2 = H(alpha u_1 + beta u_2)$$ for any scalar value $alpha$, $beta$. Your driven harmonic oscillator is currently described in such a way the combination of the force input $F(t)$ and the position $x(t)$ equals 0. As such, you could describe the system as the following: $$ 0 = H(F(t), x(t))$$ Apply the theorem of linearity on this function: $$0 = H(alpha F_1 + beta F_2, alpha x_1 + beta x_2)$$ $$0 = frac{d^2}{dt^2}(alpha x_1 + beta x_2) + 2zetaomega_0frac{d}{dt}(alpha x_1 + beta x_2) + omega_0^2(alpha x_1 + beta x_2) - frac{alpha F_1 + beta F_2}{m}$$ $$0 = alphaleft(frac{d^2x_1}{dt^2} + 2zetaomega_0frac{dx_1}{dt} + omega_0^2x_1 - frac{F_1}{m}right) + betaleft(frac{d^2x_2}{dt^2} + 2zetaomega_0frac{dx_2}{dt} + omega_0^2x_2 - frac{F_2}{m}right)$$ $$ 0 = alpha H(F_1, x_1) + beta H(F_2, x_2)$$ Because it is not intentional to have both the position and the force as inputs, the system could be described as the following: $$mfrac{d^2x}{dt^2} + 2zetaomega_0mfrac{dx}{dt} + omega_0^2mx = F(t)$$ In this case, the position is the input of the system and the force the output. If you apply the properties of linearity on this system you will notice it is linear as well.

If a constant offset in a system that is logically not really an input (for instance its not controllable), the system is indeed not linear. For example: $$y(t) = F(x(t)) = cx(t) + b$$ This is often solved by linearizing the system such that the input takes this offset into account (supposing its known or can be approximated): $$y(t) = F_l(u(t)) = cu(t), quad u(t) = x(t) + frac{b}{c}$$ With this, the function is still not linear in $x(t)$, but is linear in $u(t)$.