Kinematics in moving reference frames

Assume we have an object moving along a path $p_W(t)$ that is described in some fixed reference frame $W$. If we now have a second reference frame $B$ which differs from $W$ by some time varying rotation $R_{BW}(t)$ we can describe the objects path in the reference frame of $B$ then as $p_B(t)=R_{BW}(t)p_W(t)$. If I now want to look at the velocity of this object, we can in $W$ simply compute it as $v_W(t)=dot{p}_W(t)$. But what is its velocity $v_B(t)$ in the frame $B$? Intuitively I would say it should be $v_B(t)=R_{BW}(t)dot{p}_W(t)$, however from a mathematical perspective it would rather make sense for it to be $v_B(t)=dot{R}_{BW}(t)p_W(t)+R_{BW}(t)dot{p}_W(t)$. Which of those two are correct now, or rather what is the difference between them?

If I try to calculate the above formulas with the following example:

$$p_W(t)= begin{pmatrix} text{cos}(t) text{sin}(t) end{pmatrix},R_{BW}(t)= begin{pmatrix} -text{sin}(t) & text{cos}(t) -text{cos}(t) & -text{sin}(t) end{pmatrix},$$
so the object has a circular path and the x-Axis of frame $B$ is always along the tangent of this path in the direction of the movement. For the first formula I then get a constant velocity $v_B(t)= begin{pmatrix} 1 0 end{pmatrix}$, and in the second case a constant zero velocity $v_B(t)= begin{pmatrix} 0 0 end{pmatrix}$. Both of them seem correct, the object is indeed moving at constant speed 1 along the x-Axis of frame B. However also from the perspective of frame $B$ the object doesn’t seem to move at all, it in fact always stays at the position $ p_B(t)=begin{pmatrix} 0 -1 end{pmatrix}$.

So what is now the difference between those two velocities, both of them are described from frame $B$, however one of them actually describes a moving body, the other one a static one? I’m not actually coming from the field of Physics, so I am unfortunately not very familiar with motions in a moving coordinate frame.