Regarding the storage/cache directory... mine has folders in it with names like "61" and "c3", and they have similarly named items in them. Do we clear out all of those things leaving an entirely empty /cache directory or is it just the .js files in the root of /cache?
And here is a screenshot of the issues people keep seeing when trying to Sign up with Twitter extension...

I've tried it with and without a callback URL, but it doesn't change anything.
By comparison, I have the GitHub sign-up working, and all I needed to do there was remove the faulty callback URL and leave the field blank. After that no issues. Twitter is not playing so well.