Authors get compensated when their books are lent out through the Kindle Direct Publishing system too:

2.3 KDP Select Fund. We will establish a fund on a monthly basis and you will earn a share of that fund for each of your Digital Books included in the Kindle Owners’ Lending Library Program.

https://kdp.amazon.com/self-publishing/help?ie=UTF8&topi...

LendInk wasn't about the Kindle Lending Library though: those are books that can be borrowed without ever being purchased first. These LendInk borrows were between a book buyer and a third party, fascilitated by, say, Amazon, but without additional financial compensation for authors. As an author I still think they're a good idea to allow, but then I understand technology unlike too many other authors.

If my understanding is correct, the KDP Select Fund is for the 'Kindle Owners’ Lending Library' program, where Amazon will loan one book per month to Amazon Prime members.

LendInk facilitated lends under the 'Lending for Kindle' program, allowing a one-time lend between Kindle owners.

https://kdp.amazon.com/self-publishing/help?topicId=A2JGI9S4...

PLR seems to be mostly a European thing.