What they're "alluding to" is a standard combinator that has existed under the name "paramorphism" for a long time. Here is Jeremy Gibbons's "Origami programming" https://www.cs.ox.ac.uk/jeremy.gibbons/publications/origami.... from 2003:

    [...] capture this modified recursion pattern explicitly as a higher-order operator; in this case, the operator is known as a paramorphism [92]. In the case of lists, we could define
    
    paraL :: (α → (List α, β) → β) → β → List α → β
    paraL f e Nil = e
    paraL f e (Cons x xs) = f x (xs, paraL f e xs)

I don't know of an older source for this particular syntactic presentation, though apparently if you put in the work to translate Squiggol to Haskell, you can read this concrete syntax out of Meijer et al.'s Bananas paper from 1991.

I think what you're criticising is that the name "paramorphism" was adopted from category-theory-on-a-blackboard to functional-programming-that-actually-runs-on-a-computer. That ship sailed 30 years ago; this paper from 2024 is not what caused it to sail. I also think that you're incorrect to criticize this. Analogies are often very helpful.

You are right that the people who pointed you to Meertens's paper were not helpful in clearing up the confusion, since your confusion is exactly that Meertens's paper talks about the other meaning of the term "paramorphism".