Web is short for World Wide Web. It should therefore be capitalized. If we accept that Web is a proper noun (and I do), it cannot be arbitrary joined at the hip to the word "site." Hence, neither "website" nor "Website" is acceptable. This is consistent with most authorities I have consulted, though perhaps not with much of geekdom, which invented these words. Language changes, and nowhere is it changing more quickly than in cyberspace, but it is best that it change slowly when it comes to smashing words together. As one of my authorities points out, "if the Internet is replaced by direct-broadcast-to-the-brain technology tomorrow, website will soon look as silly as draftdodging and braburning and goldfishswallowing." (Bill Walsh, Lapsing into a Comma)
My own preference is to avoid the word "Web" altogether because it lacks precision. Not everything that is available via the Internet is on the Web. I use "Internet" or "Internet site." I especially dislike the notion that something can be "posted on the Web." I prefer to "make it available on the Bank's Internet site" or "on the Bank's internal network" or similar locutions.
Similarly, I use home page and Web page when referring to a particular virtual page, but I use "document" or "file" when I am referring to a file that can be opened from that page.
Finally, I am not an editor who insists that the language stand still to suit my prescriptions. At some point in the future I may need to change my mind about this stylistic point.