How did the 1st chiro do the c1,c2 adjusting?
Did he do or look at any xrays first?
Did he do other adjustments also?

I've had atlas/axis adjusted and it is a very gentle & non aggressive adjustment.

Have you looked at youtube videos to see how it is done, to compare?

If at the new chiro you feel very nervous , tell him what happened at the other place, maybe he can reassure you that it won't happen this time.