How are cyclic hormones characterized?
If you look in the results section of this paper on somatostatin-14, they explain that s-14 "is a fourteen residue peptide hormone where Cys$^3$ is disulfide bonded with Cys$^{14}$ to form a cyclic structure.

So your inclination is correct. The structure is cyclic and it is the cyclic structure which is responsible for the terminology. I don't know if there are exactly analogous structures with other than disulfide bonds but they would be cyclic for the same reason.

Another paper at page S1 talks briefly about the linear version of the same hormone, contrasting it with the cyclic form, again confirming that it is the structure that gives rise to the name, and not some other property.

It is the second reference that I think also answers the broader question: 'cyclic' is generic usage to describe a (simple or) complicated molecule containing a cyclic structure of joined atoms. Yet another paper (abstract) talks about cyclic vs. branched in a general way with respect to somatostatin and others:

"The aim of this contribution is to offer an unambiguous and general nomenclature system that enables researchers to represent all cyclic and branched homo- and hetero-detic peptides in a coherent manner in one-line text - as long as their as constituents can be represented in (three)-letter codes."  

In this somewhat dated paper containing proposed IUPAC nomenclature for cyclic organic compounds, in cases in which there is a question whether a molecule is cyclic or not the prefix "cyclo-" enters the formal name. But in the title the origin of the issue is clear: "Nomenclature for cyclic organic compounds..." and again confirms that the usage is not based on the elements bonded but the ringlike structure.

There is a paper in which 'heterocyclic' (vs. 'carbocyclic') is used as a subclass of cyclic to refer to rings containing other than carbon. But the term cyclic applies to both.  

Finally there is discussion of 'cyclic peptides' (wiki entry) generally which are classified 'according to the types of bonds that comprise the ring.' But they are cyclic because of the circular molecular structure notwithstanding considerable variation in the atoms (or multiplicity of bonds) comprising the ring.

Biochemistry has inherited much nomenclature from organic chemistry, and informal usage in textbook organic chemistry is that ringlike structures are called 'cyclic' whether they are comprised entirely of carbon or contain another element (typically N,O,S, etc.); for example 'cyclic ethers'--oxacyclopentane is a cyclic molecule containing oxygen.