How can an IAR ODE system be dynamic and LTI at the same time?
Your definition of time invariant is incorrect, because a time invariant system can change in time.  

Time invariant means that if a time-delay is applied on the input \$x(t+\delta )\$, it directly equates to a time-delay of the output \$y(t+\delta )\$.

Example:
When I charge a capacitor with a  constant voltage source and constant resistance in series, the voltage is time dependent.
It is "an Initially at rest ODE system" and "dynamic (time dependent)".
After e.g. 2 seconds, the voltage is x V. 

But (with the same initial conditions) it doesn't not matter when I start charging this capacitor. The system is invariant to the (start)point in time when it is "running".  

If I charge the capacitor tomorrow, after the same 2 seconds, the voltage will be same x V.

This system is also LTI.