I'm confused on some basic constructions/facts with Tate/affinoid algebras. Let $k$ be a complete non-archimedean field with abs value $| \cdot |$
$T_n = k\langle{X_1,\ldots,X_n} \rangle = \{ \sum_{J \geq 0} a_JX^{J}: |a_J| \to 0 \ \textrm{as} \ J \to \infty \}$ $J$ a multi-index is the Tate algebra over $k$ in the variables $X_1,\ldots,X_n$. This is a $k$-Banach alg with Gauss norm. It's mspec recovers the closed unit polydisc $\{(c_1,\ldots,c_n) \in \bar{k}^n |c_i| \leq 1 \}$ (modulo $\operatorname{Aut(\bar{k}/k)}$ action.)
An affinoid algebra is a quotient of the form $T_n/I$, given the quotient Banach norm/topology. I know it is a fact that two different presentations may induce different norms but they will induce the same topology.
Let $M(A)$ be max-spec of $A$. I don't understand some things regarding the handling of some subtleties of constructions like $A \langle X \rangle / (X-a) = A \langle a \rangle $ versus $A$. The former which by definition "makes $a$ power bounded" and moreover the natural map $M(A\langle a \rangle ) \subset M(A)$ naturally has image $\{ s \in M(A): |a(s)| \leq 1 \}$ where $a(s)$ means the image of $a$ in the residue field $A/\mathfrak{m}_{s}$.
But in the example $k = \mathbb{Q}_p$ and $T_2 \cong k\langle x,y \rangle$ and consider the quotient onto $k\langle x,y \rangle / (y-\frac{1}{p}) = (k\langle x \rangle) \langle \frac{1}{p} \rangle$. My question is, what is this object? If it is supposed to be the subset of $\operatorname{mspec}({k\langle x \rangle}) = \{ |c| \leq 1 \}$ such that $\frac{1}{p} \equiv \frac{1}{p} \mod (x-c)$ is power bounded then isn't this set empty? That conclusion doesn't seen correct. Any comments on what I'm getting wrong are appreciated. How are we "forcing $\frac{1}{p}$ to be power bounded", if that is possible? All definitions and notations are following lectures 1 and 2 in Brian Conrad's notes "several approaches to non-archimedian geometry".
$\mathbf{Edit}$: as reuns points out, there is no contradiction: $y-\frac{1}{p}$ is a unit in $T_2$ and so doesn't generate an ideal.