Consider some theory $T$ that is a consistent, axiomatizable extension of $\mathbf{Q}$. By the first incompleteness theorem, $T$ is incomplete, which means there is some true sentence $A$ that is not a theorem in $T$ or is not provable in $T$.
One example is the Rosser sentence:
\begin{align*} \vdash_T R_T &\leftrightarrow \forall y ( \text{Prf}_T ( \ulcorner R_T \urcorner, y) \to \exists z < y \text{Disprf}_T(\ulcorner R_T \urcorner, z)) \\ \end{align*}
Is this understanding right:
The soundness and completeness theorems say that syntactically derivable and semantically consequential are equivalent. The theorems in $T$ are the set of all sentences that are either syntactically derivable or semantically a consequence of the axioms.
We can informally reason that the Rosser sentence is true, by which we mean that any interpretation that satisfies the axioms of $T$ will assign a positive truth value to the sentence $R_T$.
But there can't exist a formal syntactical proof of $R_T$. And $R_T$ isn't a formal semantic consequence of the axioms of $T$.
Is this right?
This isn't quite right. The issue is the paragraph beginning "we can informally reason." Since $R_T$ is not a formal semantic consequence of $T$, there are interpretations satisfying $T$ which make $R_T$ false. The sense in which $R_T$ is considered true is with respect to a particular interpretation, namely the standard model of arithmetic $\mathbb{N}$. (For simplicity I'm adopting a "naive Platonist" position here, that "the natural numbers" properly refers unambiguously to something.)
That is, we have three separate notions floating around here:
Syntactic consequence, $T\vdash\varphi$ (there is a deduction of $\varphi$ from $T$ according to the particular proof system we're considering).
Semantic consequence, $T\models\varphi$ ($\varphi$ is true in every model of $T$).
Truth in a particular model of interest, $M\models\varphi$ (here $M=\mathbb{N}$).
Note that this last one makes no reference to $T$ itself; it's about a particular structure, not a theory. (Meanwhile, per the soundness/completeness theorems the first two notions coincide.)