Let me start some discussion.
If you'd like some categories come terminology to talk about what this is and what it is not, the look at this part of the LSJ entry for παράθεσις.
παρά-θεσις , εως, ἡ,
I.4.2. Gramm., juxtaposition, opp. com-position (σύνθεσις), as in Διόσ-κοροι, opp. Διο-γενής, EM278.25,649.14; also, addition, παράθεσις προθετική addition of prepositions, A.D. Synt.333.7, cf. Pron.23.12, al.
The difference there between Διὸς and Διο- is that Διός is a full noun with numbercase (genitive singular) ending suffixed to another word which it probably originally existed in a phrase with that case, while Διο- is the word-root without the addition of the endings does spell out th relationship between it and the word it is suffixed to.
Another thing that is probably going to help you is differentiation between earlier and later examples. Smyth us probably making his statement about a feature of the language that exists in the Greek of the earliest periods. The examples in that LSJ entry might suffice. The quite obvious fact is that there are far more compound words in later Greek, than there are in the classical period. Those, however, are formed as lexicosemantic additions /joinings - meaning of the roots are combined to form a new meaning, and which doesn't necessarily talk about tbe relationship between them.
A word (albiet poorly attested) like ψαρομαχία "battle of the starlings" doesn't make it clear whether the starlings were fighting or being fought against. Without making it clear, we are left to understand whether the word means fought or fought against by other means, or an authour may have felt free to use it in either sense. For a compound word Βατραχομυομαχία for example, even though doesn't show the syntactic relationship between the frog, the mouse and the battle by use of endings, the order from back to front does imply that it is a battle by the mice against the frogs. Basically μαχία is a battle. A little more complexly μυομαχία is a battle involving mice. Finally, Βατραχομυομαχία is a battle involving mice, which in turn comes to involve frogs, ie the mice make war on the frogs.
Satyrs with glass beards should not throw parties.