Here in NZ we learnt it at school and played it much the same way.
I'm not sure if we played it at Brownies as well but at GLB (Girl's Life Brigade) we didn't.

Our verses were:
Look who is here Punchinello funny fellow,
Look who is here Punchinello funny man.
(sung as each new child enters the circle)

What can you do Punchinello..........
(child in centre does an action)

We'll do it too Punchinello..........
(everyone does the action)

Who do you choose..............
(child in centre closes eyes, points and turns around and when the singing stops, the person being pointed to becomes the next Punchinello)

I'm guessing it was taught to student teachers at Teacher's Training Colleges, for it to be so widely known and so similar all around the world. That's how 'Cookie Jar' arrived in NZ.
Robyn