I would guess that the “spine” that they are referring to is the thoracic, lumbar and sacral bones, which would be 12 + 5 + 1 = 18.
The distinction between where the cervical spine ends and the thoracic spine finishes is of little to no importance because there are always only 12 thoracic vertebrae, but there can be an additional cervical spine vertebra. There are people who have six lumbar vertebrae and those whose sacrum is fused to the last lumbar vertebra, but those are exceptions to the rule. So I think the cervical spine is the neck and the spine is the rest. The coccyx, while technically a vertebra(e) is not necessarily part of the “spine”.
I don’t think you have to assume that a mistake was made because we “didn’t know much back in those days”. Chazal really knew everything, but on our level we have a hard time figuring out what they meant, so even though we seemingly know a lot, we probably have to know more details now because of our low level of understanding so things have to be clearer for us.