In most of these contexts, OWL is simply used as an acronym (such as for Online Writing Lab). However, this has nothing to do with owls per se, but rather to do with the fact that people tend to want to use acronym's without being particularly clever about it.
The most important constituent parts of the name are Online/ Oral and Writing, and the easiest jump to make from that is to stick a word beginning with L at the end. So Lab seems to be the obvious choice, despite the fact that an actual laboratory is likely one of the worst places to be for writing at length.
The reason for using the acronym OWL as opposed to something like WOW (Writing Online Workshop) is that due to the markings on the faces of many owl species, as well as their concave face and large eyes, they sometimes appear to be wearing glasses. As people wearing glasses are often associated with intelligence and academia, thus owls were often perceived to be more intelligent and wise, and associated with book reading and writing.
However, the term Owl's Nest in libraries likely extends from the fact that most libraries would have the majority of the floor space on the ground floor occupied by shelves containing books (perhaps with some seating) and an upper level or mezzanine overlooking the lower level that would have more seating and fewer books.
Hence the higher level overlooking a lower level, like looking at the ground from the height of a tree, as well as the tendency for people interested in books to spend many hours sitting stationary whilst reading, similar to a bird sitting in a nest warming their eggs, made the term 'nest' an ideal description to refer to this area. I'm not sure if this is still the case, perhaps this term has since extended to mean any area that is designated for reading books.
Again, owls were chosen as opposed to another species of bird that does the exact same thing is due to their perceived high intelligence levels thanks to the appearance of the faces of certain owl species.
Fun fact: Owls are actually one of the less intelligent species of bird.
So in summary, the two owl references in these seemingly unconnected instances of writing are just that: unconnected. The reasoning behind the two terms stem from separate unrelated locations, and in both cases the least important part of the name is 'owl', which was only chosen because of people assuming owls are clever.