The two following definitions appear in the literature.
Definition 1. A partition of a set $X$ is a set of non-empty subsets of $X$ such that every element $x$ in $X$ is in exactly one of these subsets (i.e. $X$ is a disjoint union of the subsets).
Definition 2. A partition of the interval $[a,b]$ is a non-empty finite subset $P = \{x_0,x_1,x_2,...,x_n\}$ of elements of $[a,b]$, where $a=x_0<x_1<x_2<...<x_n=b$.
These definitions do not seem equivalent, and they are confusing to me. The elements of $P$ are not even subsets of $[a,b]$. Is there some form of inconsistency?
These are different notions. The best connection between these two is obtained by looking at the intervals $[x_{i-1},x_i)$. These form a partition of $[a,b )$ according to the first definition. You wiil have to ignore one of the end points to get a partition according to the first definition.