Personally I do not think that 'log in' or 'sign in' are really satisfactory. The phrases are 'verbally skuomorphic'.
On the one hand you have 'log in' and that conjures up images of a mainframe computer with blinking lights (or something left in a toilet bowl), then, on the other hand there is 'sign in', that is a bit namby-pamby and reminiscent of what you do when you sign in when visiting someone in some posh office building.
We need to introduce a new word, something that has no real-world metaphor, that combines all the delights that go with made-up-words-for-programming. Recursive backronyms are great, plus the English language needs a word that rhymes with orange. There could be some real meaning to the word, as in, once '[xxx]-ed in' then your communications are certified NSA proof. Any suggestions?
With the right word we can introduce it in the programming community and, from there, it could enter into the wider audience and get into the OED.
'Sign in' is a relatively new contrivance, does anyone know what the first website was to use that? How did they invent it and get others to follow their example?