I did a little research and think I finally understand what zeugma and syllepsis are, and how they relate to each other.
Zeugma is any case where a single mention of a word is treated as a part of more than one clause of a sentence. Syllepsis is a type of zeugma where the word in question is used in contexts that require it to do different things.
So “He prefers dogs, she cats” is zeugma because “prefers” gets re-used, but it is not syllepsis because “prefers” is doing the exact same thing in “He prefers dogs” and the implied “She prefers cats.”
There are two types of syllepsis.
Grammatical syllepsis is where the difference is in verb form. So “I prefer dogs, she cats” would be grammatical syllepsis, because “she prefer cats” is ungrammatical – the implicit extra verb is “prefers”, a different form of the same word.
Semantic syllepsis is where the difference is in meaning. So in “And he said as he hastened to put out the cat, the wine, his cigar and the lamps”, “put out” can mean to expel, to retrieve from storage, to extinguish, and possibly to turn off (I am not sure whether that last sense would be anachronistic).
You can’t have a syllepsis without a Zeugma because the different usages have to be attributed to the same single mention, which is the definition of zeugma. So if the line from the old song went, “And he said as he hastened to put out the cat, put out the wine, put out his cigar and put out the lamps”, we would have a weird phrasing, but nothing disorienting – the casual listener might not even notice the repetition, instead hearing “put out [X]” as a whole phrase. It is where we have to attribute to each clause the same mention of the verb that our attention is called to its different meanings.
Zeugma in itself may be nothing more than parsimonious phrasing, and asylleptic zeugma may easily pass unnoticed. Syllepsis is a distinct type of zeugma because it is unexpected, and draws attention to itself.