Both have a firing pin. What the conversion does is add the transfer bar. The transfer bar along with the stepped design of the hammer make it necessary for the trigger to be pulled, raising the transfer bar to a position allowing the hammer to impact it and transfer that impact to the firing pin. The Old Model design, that without the transfer bar, used a flat faced hammer which impacted the firing pin directly.
These are the conversion kit parts for an OM:
Old Model parts: