> There is mostly no "log" that "logging in" writes to.

Um, there is, actually.

On the etymological front, "log" was originally an actual wooden log, dragged behind a ship. It then became the book in which measurements of the log's speed were entered.

The UCLA Computer Club had a log -- a book with dated entries. Someone augmented that with a wooden log ... an intricate culture developed around this pairing. Traces of this are preserved: https://americanhistory.si.edu/collections/object/nmah_69051...

(That drawing is a pretty good representation of the actual log. And there's another memorialized pun: MUS stood for machine unit second, which was the computing center measurement unit -- the Computer Club had a quota for submitted jobs [card decks].)