For a structure $mathcal{X}=(X;…)$, say that a cardinal $kappa$ is $mathcal{X}$detectable iff there is some sentence $varphi$ in the language of $mathcal{X}$ together with a fresh unary predicate symbol $U$ such that an expansion of $mathcal{X}$ gotten by interpreting $U$ as $Asubseteq X$ satisfies $varphi$ iff $vert Avertgekappa$.
For example, $omega_1$ is $(omega_1;<)$detectable since a subset of $omega_1$ is countable iff it is bounded above. By contrast, it turns out that $omega_1$ is not $mathcal{R}=(mathbb{R};+,times)$detectable.
I’m interested in the expansion $mathcal{R}_mathbb{N}:=(mathbb{R};+,times,mathbb{N})$ of $mathcal{R}$ gotten by adding a predicate naming the natural numbers (equivalently, adding all projective functions and relations). Since we can talk about one real enumerating a list of other reals, $omega_1$ is $mathcal{R}_mathbb{N}$detectable (“there is no real enumerating all elements of $U$“). More pathologically, if $mathfrak{c}=2^omega$ is regular and there is a projective wellordering of the continuum of length $mathfrak{c}$ then $mathfrak{c}$ is $mathcal{R}_mathbb{N}$detectable. So for example it is consistent with $mathsf{ZFC}$ that $omega_2$ is $mathcal{R}_mathbb{N}$detectable.
I’m curious whether this type of situation is the only way to get $mathcal{R}_mathbb{N}$detectability past $omega_1$. There are multiple ways to make this precise, of course. At present the following two seem most natural to me:

Is it consistent with $mathsf{ZFC}$ that there are at least two distinct regular cardinals $>omega_1$ which are $mathcal{R}_mathbb{N}$detectable?

Is it consistent with $mathsf{ZFC}$ that there is a singular cardinal which is $mathcal{R}_mathbb{N}$detectable?
Note that an affirmative answer to either question requires a large continuum, namely $geomega_3$ and $geomega_{omega+1}$ respectively. Although my primary interest is in firstorder definability, I’d also be interested in answers for other logics which aren’t too powerful (e.g. $mathcal{L}_{omega_1,omega}$).