You should probably read this post. Note that it does not contradict the MO post because it is about a different kind of "definable real". To make things clear, I will repeat the key definitions here:
A definable object over a theory $S$ is given by some 1-parameter sentence $φ$ such that $S$ proves $∃!x(φ(x))$. An element $c$ in a structure $M$ is definable over $M$ (without parameters) iff there is some 1-parameter sentence $φ$ over the language of $M$ such that $c$ is the unique element in $M$ that satisfies $φ(c)$. For the more general notion of "definable from parameters" see wikipedia.
First note that an isomorphic copy of the reals is definable over ZFC (i.e. there is a $1$-parameter sentence $ρ$ over ZFC such that ZFC proves ( there is a unique $R$ such that $ρ(R)$, and furthermore this $R$ is a model of the real axioms ). So we can conservatively extend ZFC to ZFC' by adding a constant-symbol $ℝ$ and axioms stating $ρ(ℝ)$ and ( $ℝ$ is a model of the real axioms ). Then (working within a meta-system that can construct the reals) the two key theorems from my post and the MO post are:
(1) Every model $M$ of ZFC' with a copy of the true reals (i.e. there is an isomorphism from the reals to $ℝ^M$) has uncountably many members of $ℝ$ (i.e. $ℝ^M$ is uncountable) but only countably many of them are first-order definable over ZFC (since there are countably many 1-parameter sentences over ZFC).
(2) If there is a model of ZFC, then there is a model $M$ of ZFC in which every element is definable over $M$. In other words, $M$ extends directly to a model $M'$ of ZFC' in which every element of $ℝ^{M'}$ is definable over $M'$. Obviously, such an $M$ does not have the true reals (since only countably many elements in $M'$ are definable over $M'$).
The point of the MO post is that not every model of ZFC' has undefinable elements of $ℝ$. After all, if ZFC has a model at all then it has a countable model, so internal uncountability within a model says nothing about definability.
The point of my post is that if we believe ZFC to be truly foundational then we ought to also believe that it has a model with the true reals, and such a model will of course have members of $ℝ$ that are undefinable over ZFC.
For the time being, you can ignore terms like "second-order concept" and "outside the universe" as they are intuitive but not really precise. For instance, NBG could be considered like a second-order axiomatization of ZFC, where classes are the second-order sort, since each class is some subset of the 'entire universe' (hence second-order). Since a definable element in a structure is defined by a 1-parameter sentence $φ$ over the language, and $φ$ corresponds to the class $\{ x : φ(x) \}$, in that sense definability is a second-order concept.
Anyway, $V$ is the class $\{ x : x=x \}$ and $HOD$ is the class of hereditarily ordinal definable sets.
