Consider the vector space $\mathbb{R}^2$ and two bases $S=\{(1,0),(0,1)\}$ and $B=\{(2,0),(0,2)\}$. If I write $(1,0)_{B}$, then there are two possible definitions/interpretations.
First: 1 and 0 are coordinates using $B$ as the basis. So this vector is in fact $(2,0)$ using the standard basis $S$.
Second: $(1,0)$ is given relative to standard basis $S$, and $(1,0)_B$ means the new coordinates of $(1,0)$ using $B$ as the basis. So $(1,0)_B=(0.5,0)$.
Are the above two definitions both being used in literature/textbooks? In my opinion, the first one is more natural as it gives us explicit information. The second definition requires some calculation to get the new coordinates and when we write $(0.5,0)$, there is a risk a confusion since the basis is not indicated.
PS: I found some textbooks use the second definition. If this definition is widely used, then how to express coordinates relative to non-standard basis like $B$? I can't just write $(x,y)$, because by default it just means $(x,y)$ w.r.t $S$, but what I really want to mean is the vector $x(2,0)+y(0,2)$.
Let me introduce some notation. Let $V$ be a finite dimensional vector space and let $B = (v_1,\dots,v_n)$ be some ordered basis of $V$. Let $\sigma_B \colon \mathbb{F}^n \rightarrow V$ be the linear map which is determined uniquely by requiring $\sigma_B(e_i) = v_i$ where $e_i$ is the standard basis of $\mathbb{F}^n$. We have
$$ \sigma_B(a_1,\dots,a_n) = \sigma_B(a_1 e_1 + \dots + a_n e_n) = a_1 \sigma(e_1) + \dots + a_n \sigma(e_n) = a_1 v_1 + \dots + a_n v_n $$
which means that $\sigma_B(a_1,\dots,a_n)$ is the unique vector in $V$ whose coordinates with respect to $B$ are $(a_1,\dots,a_n)$. This amounts to your first interpretation (i.e $(1,0)_B = \sigma_B(1,0)$).
Your second interpretation is that the vector $[v]_B$ is the vector of coordinates of $v$ with respect to $B$ and this amounts to $[v]_B = \sigma_B^{-1}(v)$. From my experience, this is the more common notation. Namely, one uses $[v]_{B} := \sigma_B^{-1}(v)$ to denote the coordinates of $v$ with respect to the basis $B$ rather then using $[(a_1,\dots,a_n)]_B$ (which should be defined then as $\sigma_B(a_1,\dots,a_n)$) to denote the unique vector in $V$ whose coordinates with respect to $B$ are $a_1,\dots,a_n$.
Your observation amounts to the fact that it is easier to calculate $v$ given $[v]_{B} = (a_1,\dots,a_n)$ (this amounts to computing $\sigma_B$ which is given explicitly) rather than $[v]_B$ given $v$ (this amounts to computing the inverse map $\sigma_B^{-1}$ so in general you need to invert it).