Regarding potential distinctions between autonomous, non-autonomous, time-invariant and time-varying systems, I have found out opinions supporting that:
autonomous systems are time-invariant and non-autonomous are time-varying, (A treatise on stability of autonomous and non-autonomous systems, R. Adhikari - 2013)
linear systems can be time varying or invariant, nonlinear systems can be autonomous or non-autonomous Book
systems can be simultaneously characterised as any combination among the above, which indicates that they cannot be the same thing. (Adversary Control Tactics for Cyberphysical Systems, Kontouras, 2020 - non-autonomous linear time invariant) (Realization and identification of autonomous linear periodically time-varying systems, Markovsky et al.-autonomous linear time varying)
The example is about LTI but it would be the same for LTV.
This system:
$\ddot{y}-b\dot{y}-ay=u(t)$
$y=x_{1}$, $\dot{x_{1}}=x_{2}$, $\dot{x_{2}}=ax_{1}+bx_{2}+u(t)$
$\dot{\vec{x}}=\begin{bmatrix} 0 & 1\\ a&b \end{bmatrix}\vec{x}+\begin{bmatrix} 0\\ 1 \end{bmatrix}u(t)$
, is time-invariant since $A$, $B$ matrices are not time-dependent, and since there is a time-dependent input $u(t)$, it should additionally be non-autonomous.
If we feed the output $y$ back, the input $u(t)$, will now become a function of state variables. In this sense, the system should be autonomous.
$\dot{\vec{x}}=\begin{bmatrix} 0 & 1\\ a&b \end{bmatrix}\vec{x}+\begin{bmatrix} 0\\ 1 \end{bmatrix}u(t)$
$y=x_{1}$
$u(t)=-ky$
Do the previous make sense? Is it that open-loop control systems are in principle non-autonomous, while closed-loop ones are autonomous, in the absence of additional time-dependent disturbances?
