They are both correct. 1/64th is the accurate value used to scale down a prototype dimension to S scale. Everything in S scale is 1/64 full size. The O scale equivalent of this number is 1/48.
3/16" is the resulting size in S scale of a prototype one foot. The O scale equivalent of this value is 1/4". Recognize that the 3/16" value has a very slight rounding error to simplify the fraction.