Primes p such that the expansion of 1/p takes a full p - 1 digits (and not any less) to repeat are called “full reptend primes”, or, in other jargon, “primes p such that 10 is a primitive root modulo p”. There’s a 1-to-1 correspondence between them and cyclic numbers, as you began to note; the finite block of repeating digits of any full reptend prime describes a cyclic number, and conversely, repeating any cyclic number infinitely produces the decimal expansion of the reciprocal of a full reptend prime.
As far as I’m aware, there’s no particularly nice way to tell whether a prime p is or is not of this form, except by, well, checking how long it takes for the expansion of 1/p to repeat digits (equivalently, finding the smallest power of 10 which is equal to 1 modulo p). But here’s a list…