On gradient boosting and types of encodings
This is actually a feature of tree-based models in general, not just gradient boosting trees.
Not exactly a reference, but this Medium article explains why ordinal encoding is often more efficient.
On the topic of safety, I think the author should have said that the use of ordinal encoding is more safe compared to linear methods, but still not perfectly safe.  It's possible for decision-tree methods to find spurious rules within ordinal encodings, but they don't have the strong assumptions about numeric semantics that linear methods do.

. . . I would like to know more about their supposed equivalence to encode unordered categorical variables . . .

Any rule derived with one-hot encoding can also be represented with ordinal encoding, it just might take more splits.
To illustrate, suppose you have a categorical variable foo with possible values spam, ham, eggs.  A one-hot encoding would create 3 dummy variables, is_spam, is_ham, is_eggs.  Let's say an arbitrary ordinal encoding assigns spam = 1, ham = 2, and eggs = 3.
Suppose the OHE decision tree splits on is_eggs = 1.  This can be represented in the ordinal decision tree by the split foo &gt; 2.  Suppose the OHE tree splits on is_ham = 1.  The ordinal tree will require two splits: foo &gt; 1 then foo &lt; 3