I can think of two nonequivalent ways of defining an interval in a poset:
An interval of a poset $P$ is a subset $I\subset P$ with the property that for all $x, y, z\in P$ such that $x < y < z$ and $x, z\in I$, we have that $y\in I$.
An interval of a poset $P$ is a subset $I\subset P$ of one of the 9 forms: $P$, $\{\,y\in P\mid x < y\,\}$, $\{\,y\in P\mid x\le y\,\}$, $\{\,y\in P\mid y < z\,\}$, $\{\,y\in P\mid y\le z\,\}$, $\{\,y\in P\mid x < y < z\,\}$, $\{\,y\in P\mid x\le y < z\,\}$, $\{\,y\in P\mid x < y\le z\,\}$, $\{\,y\in P\mid x\le y\le z\,\}$, where $x, z\in P$.
Is there a consensus about which of these two is the "right" one? Respectable references are welcome.
I would think that the first definition is better, but it is the second that is given in Bourbaki's Theory of Sets and on nLab wiki.
An additional related question: does an interval have to be nonempty?
The first definition is fine if you are dealing with a total order, but it doesn't match my intuition for what an interval is otherwise. For example, let us take $\mathcal{P}(\mathbb{N})$ partially ordered by $\subset$.
I would not consider $\{\{2\},\{3\},\{4\}\}$ to be an interval here. But it matches Definition 1. Something like $\{\{2\},\{2,3\},\{2,679\},\{2,3,679\}\}$ makes for a much better interval.
Whether the empty set should be considered an interval or not really varies from setting to setting.
So, for interval, I'd go with Definition 2. Definition 1 describes a convex set instead.