ORIGINAL RESPONSE:
https://www.jstor.org/stable/2273786?seq=1#page_scan_tab_contents Is the article where it is from. It seems to have never been added to the library, which would be my fault.
https://projecteuclid.org/download/pdf_1/euclid.pl/1235417266 is the first chapter of the textbook where I personally found it, although I would use the JSTOR article in the library of Cantor's Attic after this post.
The massive textbook I used is called "Model-Theoretic
Logics" by the "Ω-Group" (the coolest pen-name for a group of mathematicians). In Part F, this equivalence is proven.
Chapter XVII is "Set-theoretic definability of logics" (written by Väänänen, one of my heroes) which is where this comes from. The definition is quite nuanced, but it is a great read. I recommend this textbook.
What Is a Logic? (loose definition)
We start with a "vocabulary." $\tau$ is often used for this type of object; it is defined as a (nonempty) class of constant symbols, finitary predicate symbols, and finitary function symbols (simple enough). However, you also have to have "sort symbols" for every function symbol and every constant symbol, and furthermore you have to have sort symbols for every argument of every relation symbol and every input of every function symbol.
The sort symbols are just for well-defining "terms" and their syntactics. The "$\tau$-terms" are built up of just function symbols applied to constant symbols, each of which are then given a "sort" to help define things later.
Using the sort, you can well-define the number of inputs in a given "formula" made out of $\tau$.
An "abstract logic" $\mathcal{L}$ gives every vocabulary $\tau$ a class $\mathcal{L}(\tau)$ of formulae and "atomic $\tau$-formulae of $\mathcal{L}$." More important is that it gives every $\tau$ a relation $\models^\tau$ which defines semantics: how a structure interprets a formula of $\mathcal{L}$.
The kind of logic we are talking about has been formalized by Väänänen, but it suggests that the definitions of $\models^\tau$ and deciding what things are atomic formulae are recursive somewhat (synctatic) and furthermore there is a set $A$ such that every formula of $\mathcal{L}$ is in $A$ and that $\mathcal{L}$ can easily describe this set $A$ (the syntax of $\mathcal{L}$ is representable).
There is a lot more to this definition that I haven't talked about, so this is a very rough summary. I urge you to take a look at the book instead of just taking my word (I am in no ways an expert).
Addendum:
I've recently circled back to this problem after nearly 5 years, and I now realize that the actual argument for this proof isn't as complicated as I thought it was.
For anybody reading this, here is a simple definition of a logic, so that VP is equivalent to the statement that every logic has a strong compactness cardinal.
A logic consists of two parts:
- The language, which is a proper class $\mathcal{L}$ of $\tau$-sentences for each relational signature $\tau$; these are functions $\varphi:x\rightarrow\tau$ for some set $x$. The class of $\tau$-sentences for a particular $\tau$ may be denoted $\mathcal{L}[\tau]$.
- The semantics, which is a proper class $\models_{\mathcal{L}}$ of tuples $(\mathcal{M},\varphi)$, where $\mathcal{M}$ is a $\tau$-structure for some signature $\tau$ and $\varphi\in\mathcal{L}$ is a $\tau$-sentence.
We require that it satisfies the following conditions:
- Isomorphism invariance: if $\mathcal{M}$ is isomorphic to $\mathcal{N}$, then for any $\varphi$ in $\mathcal{L}$, $$(\mathcal{M}\models_{\mathcal{L}}\varphi)\leftrightarrow(\mathcal{N}\models_{\mathcal{L}}\varphi)$$
- Reduction invariance: if $i:\tau\rightarrow\sigma$ is a monomorphism of relational structures, and $\mathcal{M}$ is a $\sigma$-structure, then for any $\varphi:x\rightarrow\tau$ in $\mathcal{L}$, the composition $i\circ\varphi:x\rightarrow\sigma$ is also in $\mathcal{L}$, and $$(\mathcal{M}|_{\tau}\models_{\mathcal{L}}\varphi)\leftrightarrow(\mathcal{M}\models_{\mathcal{L}} i\circ\varphi)$$
These properties essentially guarantee that each $\varphi$ is a statement, with parameters in $V$, about the relations in the structure; and nothing more.
Of course, in the above formulation, logics like $\mathcal{L}_{\infty,\infty}$ can be construed, which have no strong compactness cardinal no matter how you slice it. We can say that the logic is small when there is some fixed set $X$ such that each $\tau$-sentence $\varphi\in\mathcal{L}$ is a function $x\rightarrow\tau$ for some $x\in X$.
As an example, $\tau$-sentences of $\mathcal{L}_{\kappa,\kappa}$ may be construed as functions $x\rightarrow\tau$ for $x\in H_\kappa$. So, this logic is small.
The proof:
VP is equivalent to the statement that for every class $A$ there is a stationary class of $A$-extendible cardinals. So, assuming VP, there is a $\models_{\mathcal{L}}$-extendible cardinal $\kappa$ such that the set $X$ mentioned above is in $V_\kappa$.
Let $\Sigma\subset\mathcal{L}[\tau]$ be such that for every $t\subset\Sigma$ with $|t|<\kappa$, there is a $\tau$-structure $\mathcal{M}_t$ such that $\mathcal{M}_t\models_{\mathcal{L}}\varphi$ for all $\varphi\in t$. In other words, let $\Sigma$ be a $\kappa$-satisfiable $\mathcal{L}$-theory. Then, let $j:(V_\eta;\in,\models_{\mathcal{L}})\rightarrow (V_\theta;\in\models_{\mathcal{L}})$ be an elementary embedding with critical point $\kappa$, such that $\Sigma\in V_\eta$ and each $\mathcal{M}_t\in V_\eta$.
Since $V_\eta$ witnesses that every subset of $\Sigma$ of size below $\kappa$ has a model, by elementarity and $\models_{\mathcal{L}}$-correctness, $V_\theta$ witnesses that every subset of $j(\Sigma)$ of size below $j(\kappa)$ has a model. In particular, $j"\Sigma$ is a subset of $j(\Sigma)$ of size $\kappa$, so it must have a model $\mathcal{M}$. However, we are not quite done.
This $\mathcal{M}$ is a $j(\tau)$-structure, satisfying every member of $j"\Sigma$. So, we don't quite have a model of $\Sigma$ per se. However, every $\varphi\in j"\Sigma$ is of the form $j(\varphi)$ for $\varphi:x\rightarrow\tau$. Because $x\in V_\kappa$, we have that $j(\varphi)=j|_\tau\circ\varphi:x\rightarrow j(\tau)$, and $j|_\tau:\tau\rightarrow j(\tau)$ is a monomorphism of signatures.
So, by reduction invariance, $\mathcal{M}|_\tau$ is a model of every $\varphi\in\Sigma$, completing the proof.
The reverse direction is not as hard. Given a proper class $A$ of $\tau$-structures, we wish to find a first-order elementary embedding between two of its members. Because first-order logic is isomorphism invariant, WLOG $A$ is closed under isomorphism. For each monomorphism $i:\tau\rightarrow\sigma$, we can consider the class $A^i$ of $\sigma$-structures who reduce to a member of $A$. If there is an elementary embedding in $A^i$ for some $i$, then since first-order logic is reduction invariant, there is an elementary embedding in $A$.
So, we construct the logic by starting with first-order logic, adding a $\tau$-sentence satisfied only by members of $A$, and then for each monomorphism $i:\tau\rightarrow\sigma$ adding a $\sigma$-sentence satisfied only by members of $A^i$. This logic is readily seen to be isomorphism invariant and reduction invariant by the exposition above. It is also small. If it has a strong compactness cardinal, pick any structure $\mathcal{M}$ in $A$ of size greater than that of the cardinal, and consider its elementary diagram in this logic. Add to it a new constant symbol and, for each $x\in\mathcal{M}$, an axiom stating that $x$ is not equal to the new constant symbol. This theory is clearly $\kappa$-satisfiable, so by strong compactness it is satisfiable. But any model of it must be a member of $A$ admitting a nontrivial elementary embedding from $\mathcal{M}$. QED
