What are the essential criteria for sleep in living organisms?
I take it you did not even read the article. The article quite nicely expounds on this question. The whole idea is that sleep used to be considered as a human or psychological phenomenon that could only be studied by EEG, and clearly sleep was a phenomenon that goes deeper and wider in biology than just EEG waves. From the same article, just a little below:

She distilled a set of behavioral criteria to identify sleep without the EEG. A sleeping animal does not move around. It is harder to rouse than one that’s simply resting. It may take on a different pose than when awake, or it may seek out a specific location for sleep. Once awakened it behaves normally rather than sluggishly. And Tobler added a criterion of her own, drawn from her work with rats: A sleeping animal that has been disturbed will later sleep longer or more deeply than usual, a phenomenon called sleep homeostasis.

The keyword is behavioral criteria.
This is cited, I recommend a read-through.
I also recommend reading the rest of the article, it guides you through the status quo in sleep science which may further answer more of your questions. At the end, there is an exposition on animals without brains that seem to be sleeping. The hydra is one of them.