I can't imagine Birmingham, Alabama being more liberal than New York City or Boston, but I've never lived in Birmingham. I can tell you this: Providence is a hell of a lot less liberal than Boston, and was ramked as more liberal.
Providence is distressingly Democrat, not so much that it doesn't have a republican mayor and votes for a RINO Senator, though.
Providence County voted 36% for Bush. Jefferson county, dominated by Birmingham, voted 54% for Bush. Suffolk County (almost coterminous with Boston) voted only 23% for Bush, and DC only 9% for Bush. Kings County (Brooklyn) was 24% for Bush; Queens 27%, Richmond (Staten Island, NY) 56%
Re: Providence vs. Boston. Boston is not as liberal as you're painting it, and probably is similar to Providence in terms of having a big liberal and a big conservative contingent that vote Democrat together but hate each other. Boston's suburbs are what give the whole area its reputation as a liberal haven: Cambridge, Somerville, Newton, Brookline. Combined they have a population about 50% that of Boston but much more liberal.