I consulted several books and found that the definitions of faces and exposed faces of convex sets are a bit messy. Many books just treat them as the same stuff.
In book "foundations of optimization", page 128, the author give a definition of faces:
Let $C$ be a nonempty convex set in a vector space $E$. A face of $C$ is convex subset $F\subseteq C$ such that if $x,y\in C$ and the line segment $(x,y)$ intersects $F$, then $[x,y]\subseteq F$.
Then, the author give an equivalent definition:
Let $F$ and $C$ be two convex sets such that $F\subset C$. Then $F$ is a face of $C$ if and only if $C\setminus F$ is a convex set.
I doubt the two of them are equivalent. Here is a counterexample.
Consider $\mathbb{R}^2$. Let $C=\{(x_1,x_2):x_1^2+x_2^2\leq1\}$ be the unit disk and $F=\{(x_1,x_2):(x_1,x_2)\in C, x_2\geq0\}$.
By the second definition, $F$ is a face of $C$. However, $F$ is clearly not a face of $C$ by the first one.
I am not sure if there is any misunderstanding. Any help is appreciated.