Is the centre of rotation in non-inertial rotating frame different from the actual centre? (Is my animation correct) 
(Sorry for confusing question, here's the actual explanation)
If Im sitting on the edge of a rotating merry-go-round, then from my perspective the world is rotating around me. Now, there is a fixed centre around which Im 'actually' rotating from someone else's perspective.
But from my perspective, is the world rotating around that same centre as well?
OR
Is am 'I' the centre around which the world is rotating?(mind that Im sitting on 'edge')
OR
Is is somewhere between me and the actual centre?
 A: 
If I'm sitting on the edge of a rotating merry-go-round, then from my perspective the world is rotating around me.

No, it isn't. Think about this carefully (or find a merry-go-round and do a practical experiment). From your perspective the merry-go-round is stationary and the rest of the world is rotating about the centre of the merry-go-round.
Think about the point in the outside world that is nearest to you. When the merry-go-round has rotated through $180^o$ this point is now on the opposite side of you and the opposite side of the merry-go-round. If the world appeared to be rotating about you then this point  would stay the same distance away from you as the merry-go-round turned, which it clearly does not.
Therefore, whether you think of yourself as stationary and the world as rotating or vice versa, there is only one fixed centre of rotation, which is always the centre of the merry-go-round.
A: If there is a non rotating object (like a pole) in the center of the merry go round, this object will look rotating around itself, from the edge person perspective.
But all other non rotating objects (the rest of the world) will rotate around the pole in the center.
It is easy to see that for non rotating objects close to the person in the edge at some moment. Of course those object will be far away after half a cycle. So they are not rotating around the person.
A: $\def \b {\mathbf}$


In both cases the center of rotation is the same ,but you are looking at the   object towards the vector $~\mathbf u~$,   from differet perspective .
The equations
I)
$$\b u=\b r_o(\psi)-\b r(\omega\,t)$$
II)
$$\b u=\b r_o(\omega\,t)+\b r(\varphi)$$
A: I think the notion of "actual centre" depends on the frame in which you are considering your rotation motion.
For an observer outside of the rotating frame, the merry-go-round actually has a centre about which it rotates. But when we shift frames to the person on the mere-go-round, the centre in the rotating frame would be the point where all the fictitious forces which are experienced by a person in the rotating frame vanish (i.e the actual centre of rotation).
Thus the centre of the rotation is the same irrespective of who is viewing the motion, the difference is only how you describe that point.
I hope this helps.
