"All true theorems are logically equivalent" 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.
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.