Origin of "Under the weather"?

The Phrase finder provides an explanation of the origin:

: To be under the weather is to be unwell. This comes again from a maritime source. In the old days, when a sailor was unwell, he was sent down below to help his recovery, under the deck and away from the weather.

... Here's a similar one I found: "Under the weather. To feel ill. Originally it meant to feel seasick or to be adversely affected by bad weather. The term is correctly 'under the weather bow' which is a gloomy prospect; the weather bow is the side upon which all the rotten weather is blowing." From "Salty Dog Talk: The Nautical Origins of Everyday Expressions" by Bill Beavis and Richard G. McCloskey (Sheridan House, Dobbs Ferry, N.Y., 1995. First published in Great Britain, 1983).

Another site states that something similar:

Passengers aboard ships become seasick most frequently during times of rough seas and bad weather. Seasickness is caused by the constant rocking motion of the ship. Sick passengers go below deck, which provides shelter from the weather, but just as importantly the sway is not as great below deck, low on the ship.

In both cases, we have two things in common. One, is its origin came from sea travel, when people felt ill due to several reasons, and the other thing in common, is that they both cited the fact that the persons feeling unwell went below deck.

I would suggest that "Salty Dog Talk" has misrepresented the phrase, which should be "under the weather rail," which is where tired (or ill) sailors would stow themselves to get some rest, as documented in Two Years Before the Mast by Richard Dana (available online for perusal). There is no such thing as a "weather bow," as the bow is the front of the ship (or boat). As well, the bow of a heaving ship is the worst place to be, if one is feeling "under the weather."