Estimation of accelerating target using position measurements only
I don't have a really good answer for you, but I can see a possible problem with your second attempt:  the value of $a_3$ has the potential to completely blow up the dynamics because it is multiplying $t^3$.

In cases like this, you are better off defining an interval over which you wish to do the estimation, $[T_1,T_2]$ and then choosing well-behaved polynomials on this interval (e.g. the Legendre polynomials, $P_k(x)$)  and multiplying them.

That way your measurement becomes
$$
y(t) =  \alpha_1 P_0(x;T_1, T_2) + \alpha_2 P_1(x;T_1, T_2) + \alpha_3 P_2(x;T_1, T_2)
$$
where $P_k(\cdot;T_1, T_2)$ is the (scaled) $k^{\rm th}$ order Legendre polynomial:
$$
P_k(\cdot;T_1, T_2) = \frac{1}{T_2 - T_1} P_k\left( \frac{2t -T_1 - T_2}{T_2 - T_1} \right)
$$
where $P_k(\cdot)$ is the standard Legendre polynomial.

I've used the technique in a formulation of the EKF for frequency tracking.