In the Expanded Edition of A Basis Theory Primer by Christopher Heil, the following definitions are made:
Definition 1.4. A sequence $\left\{x_n\right\}$ in a Banach space $X$ is:
(a) bounded below if $\inf\left\|x_n\right\|>0$,
(b) bounded above if $\sup\left\|x_n\right\|<\infty$,
(c) normalized if $\left\|x_n\right\|=1$ for all $n$.
How are (a) and (b) motivated?
It feels like (a) has a typo because $\left\{a_n\right\}_{n\in\mathbb N}$, where $a_n=-n$, is a sequence in $\mathbb R$ that (to me) is not bounded below because $a_n\to-\infty$. However, $\inf\left|a_n\right|=1>0$. Thus, according to (a), it is bounded below.
Similarly, it feels like (b) is the definition of bounded above and below.
What am I missing here?
Short answer: this is their definition of bounded below/above, and you should just follow it when reading the book.
Intuition: bounded above in this context means "contained in some large ball of finite radius," and "bounded below" means "outside some small ball centered at zero, i.e., no subsequence converging to zero." It is indeed different with the particular notion of bounded above and below in $\mathbb{R}$ that you mentioned. Once you recognize this difference, you should just follow their definition and not let it bother you too much.