Acceleration in polar coordinates and fictitious forces

On the one hand, we can express the acceleration $vec{a}$ of some mass in polar coordinates as:
$$ vec{ddot{r}} = ({ddot{r}}-rdot{theta}^2)hat{r} + (rddot{theta}+2dot{r}{dot{theta}})hat{theta}$$
On the other hand, from the reference frame of the particle, we can say:

$$vec{ddot{r}} = frac{vec{F}}{m}+2vec{dot{r}}timesvec{omega}+(vec{omega}timesvec{r})timesvec{omega} – dot{vec{omega}}times vec{r}$$

I was wondering what is the exact connection between the two equations, obviously that can’t be a coincidence that they are very much similar (the Coriolis force, the Euler force etc).

The first equation is from an inertial perspective (correct me if I’m wrong), while the latter is non-inertial. In the first equation, the mass has an inward force due to its rotation, while in the latter the force is outward. Are all the changes just a matter of perspective?