I've seen the phrase "all true theorems are logically equivalent" thrown around here, when people ask if a theorem X and a theorem Y are logically equivalent.
What is meant by this? Are they just referring to the fact that an implication with a true consequent is always a true statement, therefore the biconditional $P \Longleftrightarrow Q$ is true for any true $P$, $Q$, or do they mean something more meaningful?
Yes, that must be what people who throw around that phrase mean: If $P$ and $Q$ are both true (or provable in some particular theory), then $P\Leftrightarrow Q$ is also true (or provable in that theory).
However, this is not actually what "logically equivalent" means in logic. The usual meaning of that is that $P$ and $Q$ are logically equivalent if and only if $P$ has the same truth value as $Q$ in every interpretation. Or, equivalently, $P$ and $Q$ are logically equivalent if $P\Leftrightarrow Q$ is provable without using any non-logical axioms.
One can speak about being equivalent relative to some theory -- for example the Axiom of Choice and Zorn's Lemma are equivalent relative to ZF (or"given ZF"), which simply means that ZF proves AC$\Leftrightarrow$Zorn. People often just say that "AC and Zorn are equivalent", in which case they are leaving which theory they are talking about implicit. Usually it is clear from the context what the underlying theory is.
The word "logically" should not be used in the latter case, though.
I would use such a phrase to castigate someone who asked whether two theorems were logically equivalent. There is no such formal relationship.
What is actually discussed sometimes is whether theorems can be easily derived from each other, using techniques that are much simpler than the original proofs. This is a human thing, the judgment of "simpler" is in the eye of the beholder. Indeed, human judgment is exactly what happens when some result is called a Corollary of another.
One example: I am not sure about full generality, but in the case where all curves allowed are piecewise analytic, the three theorems (1) Jordan Curve Theorem (2) the graph $K_5$ is not planar (3) the graph $K_{3,3}$ is not planar, all follow from each other quickly.
In some cases, when someone says "$P$ and $Q$ are equivalent", they mean more than just "$P \Longleftrightarrow Q$ is provable". They mean that a proof of $P \Longleftrightarrow Q$ is much easier than a proof of $P$ or a proof of $Q$.
The motto "All true theorems are logically equivalent" is in opposition to this.
No, that's a statement derived purely from the definition of equivalence There's no deep meaning there.
This may be true, but is useless in helping you to prove anything.
It is, as I read somewhere years ago, "a theorem so general that it has no particular application."
All theorems are technically indeed equivalent—though not necessarily logically equivalent—to one another.
However, informally, the assertion "statements (or, sometimes, theorems) P and Q are equivalent" is generally intended to mean that they are easily derivable from each other.
The only logical component of a statement is whether it is true or false. That is logic. So saying they are logically equivalent means that they are either both true or both false.
We may not know whether they are true or false, but we may still be able to show that there truth values coincide. In other situations, the truth or false of the statement "depends" on context (variables). In this case, logical equivalence means they have the same truthiness in every scenario.
