> Puzzle 1: what is a "poset" according to Chapter 1 of Fong and Spivak's book?

They define a poset as a set with a [preorder](https://en.wikipedia.org/wiki/Preorder).

> Puzzle 2: how does their definition differ from the usual definition found in, e.g., Wikipedia or the nLab?

Ordinarily, a poset is a set with a [*partial order*](https://en.wikipedia.org/wiki/Partially_ordered_set#Formal_definition) rather than a preorder.

A partial order satisfies an *anti symmetry* axiom: if \$$x \preceq y \$$ and \$$y \preceq x \$$ then \$$x = y \$$.

> Puzzle 3: what do mathematicians usually call the thing that Fong and Spivak call a poset?

They are usually called [preordered sets](https://en.wikipedia.org/wiki/Category_of_preordered_sets).

Modal logicians call them "[S4 Kripke Frames](https://ncatlab.org/nlab/show/the+logic+S4%28m%29#relation_to_kripke_frames)".

> Puzzle 4: list some interesting and important examples of posets that haven't already been listed in other comments in this thread.

Java classes form a *preordered set* when ordered by inheritance.

Natural numbers \$$\mathbb{N}\$$ form a *traditional poset* under the relation "\$$x\$$ divides \$$y\$$", often denoted "\$$x\ |\ y\$$".

For two sets \$$A,B \subseteq \mathbb{N} \$$, we say \$$A\$$ *is Turing reducible to* \$$B\$$ if there is an [oracle machine](https://en.wikipedia.org/wiki/Oracle_machine) with oracle tape containing \$$B\$$ that can compute the characteristic of \$$A\$$. This is denoted \$$A \leq_T B\$$. Under \$$\leq_T\$$, \$$\mathcal{P}(\mathbb{N})\$$ is a *preordered set*.

EDIT: Cleaning up terminology