I must note that the notion of lenses meant by this paper is very different from what is commonly called lenses nowadays. The modern definition of a lens is a function p a (f b) -> p s (f t) where p is a profunctor and f is a functor. It's the functional programming equivalent of getters and setters, but composable.
When you say "the modern definition of X is Y", it tends to suggest that the older definition is no longer relevant or useful, even though I know that's not what you mean. So to clarify for everyone here, and to reinforce your first sentence, these are just unrelated concepts that happen to use the same term "lens". The lenses in this paper are still as relevant as ever, but these days they are usually called "unfold" (or "anamorphism" for the category theorists).
It’s an abstract concept that you can implement in many different ways.