the child moves, the chair doesn't:
rocking around on the chair
rocking back and forth on the chair
squirming / wiggling / fidgeting
not sitting still

the child makes the chair move up and down or from side to side:
rocking the chair
jiggling the chair
tipping the chair
playing with the chair
moving the chair around

the child makes the chair move so that the front legs of the chair are not on the floor:
rocking back in the chair
tilting the chair backwards
leaning the chair back



'Rocking with the chair' sounds like a bit of a Germanism to me. If you rocked with a chair, you might be beside it, imitating its action, pretending you were also a rocking chair, doing the same thing along with it.

to tilt on one's chair, the child tilts on the chair, Don't tilt on the chair!
Isses das?