So here’s what I’ve found (so far), Mark.
‘Spyder’ is a bastardisation of the English ‘spider’. Apparently, an Irish carriage builder in the 19th century named one of his models the ‘Spyder’ due to its web-like structure (example images here:
http://www.coyaltix.com/spyder.html) and his being somewhat subversive of the English, hence ‘y’, not ‘i’. I wasn’t able to verify this, but sounds likely.
The explanation here
http://www.veloce.se/WELCOME.HTM#spider is that ‘spyder’ was corrected to ‘spider’ by the Italian National Federation of Bodymakers at a conference in Milan in 1924. I haven’t been able to verify anything about the organisation nor the conference, but Italy (Europe as a whole) was highly unionised at the time, so it makes sense that something like that might have happened.
There were Spiders built much earlier than Porsche’s 550 or 356. In fact, here’s a photo of a 1924 Fiat Corsa Spider
which I found here:
http://www.madle.org/og05fiat501srace1924.jpg. Apparently, there were Alfa Romeo Spiders and Rio Spiders from around the same time, so that could (tenuously) point to the 1924 conference.
Sedan, phaeton, and brougham were also names for horse carriages brought over into autos, so it's logical that the word came from the days of horse-and-buggy and was carried over by the carriage makers during the time they ventured into autobody building.
Unless anyone's planning a visit to Milan's archives with an Italian-to-English translator, that's all I have.