'Third wheel' or 'fifth wheel'?
If you are the "extra" person in a situation, are you the "third wheel" or the "fifth wheel"?
Some books—like Film Noir Guide—say "third":
O'Keefe plays an escaped convict on the run with his moll (Trevor). Along the way they grab a hostage (Hunt) who works for O'Keefe's lawyer. O'Keefe and Hunt eventually fall for each other, leaving the jealous Trevor feeling like a third wheel. Burr is the hood O'Keefe took the rap for, and Ireland is Burr's vicious henchman. The cast in this hard-boiled and fast-paced noir is excellent, especially Trevor as the lovesick tag-a-long.
but others—like A Wife in Musashino—say "five":
Tomiko was especially kind to her daughter this evening. She held Yukiko's hand, stroked her hair, and explained various things about the children's story Tsutomu was narrating. She enjoyed demonstrating in front of her rival an affection she held in common with Tsutomu.
Tomiko's arrow struck home. Michiko felt like a fifth wheel sitting next to her. Her wariness regarding Tomiko's flirtations, which she had first experienced when she wasn't sure of her feelings for Tsutomu, came back to her.
Solution 1:
The traditional idiom
Christine Ammer, The American Heritage Dictionary of Idioms (1997) has this entry for "fifth wheel":
fifth wheel An extra and unnecessary person or thing, as in He was the only one without a date, so he felt like a fifth wheel. This expression, which alludes to an unneeded wheel on a four-wheel vehicle, may have originated as long ago as 1631, when Thomas Dekker wrote Match Me in London: "Thou tiest but wings to to a swift grey Hounds heel, And addest to a running Chariot a fifth wheel."
A Google Books search confirms Ammer's assertion that the original allusion is to an unnecessary wheel added to a four-wheeled vehicle. In Google Books search results for the phrases "like a/the fifth wheel" and "like a/the third wheel," most of the earliest matches involve fifth wheels. For example, from "Review of New [missionary] Publications," in The Quarterly Christian Spectator, volume 6 (May 1824):
In glancing our eye over the country, we cannot help receiving the perhaps erroneous impression that in relation to the great system, some of these institutions are too much like the fifth wheel to a coach.
From 'A Woman,' England, the Civilizer: Her History Developed in Its Principles (1848):
Vice pollution, luxury, misery, vanity, deforming the face of society; and indigence stealing on the land preparing for pauperism. In Ireland the sword of desolation. In Scotland the canker of corruption. One chained to the car of the victor. The other sold by her children. £20,000 distributed among the members of Scotland's assembly fastened that ancient realm like a fifth wheel to the state carriage of England.
From a debate on the Pardoning Power (June 20, 1853), in Official Report of the Debates and Proceedings in the State Convention, Assembled May 4th, 1853, to Revise and Amend the Constitution of the Commonwealth of Massachusetts (1853):
Mr. CROWNINSHIELD. ... I believe the true remedy is the one suggested by the gentleman from Milton, (Mr. Churchill,) because it does not enter into details which ought not to be incorporated into the Constitution. I believe the true remedy is to be found in abolishing the council, for I do think that it is like a fifth wheel to a coach, entirely unnecessary in what is called the glorious 19th century, and for which there is no occasion, and I am decidedly inclined to the opinion that the government can be better administered without than with it.
From comments of Eloi Rioux, mayor of Trois Pistoles," in Appendix to the Fifteenth Volume of the Journals of the Legislative Assembly of the Province of Canada (1857):
2nd. The office of County Superintendent, besides being very burthensome, appears like the fifth wheel to a carriage, that is to say, if not injurious, it is at least useless.
And from Bayard Taylor, The Story of Kennett (1866):
"But there must be some other way," Gilbert insisted. "Do you never walk out together? And couldn't we arrange a time — you, too, Betsy, I want to see you as well!"
I'm afeard I'd be like the fifth wheel to a wagon."
"No, no! You must be there — you must hear a good part of what I have to say."
Nevertheless, there is one striking example of "like a third wheel" from this early period. From Alexander Ross, Adventures of the First Settlers on the Oregon or Columbia River (1849):
Mr. Clarke also accompanied the North-West brigade, on his way to Astoria. With the craft peculiar to Indian traders, they had crammed down Mr. Clarke's throat that nothing could be done at Astoria without him, although his accompanying them was like the third wheel to a cart; but it answered their purpose: for his leaving Spokane threw at once all the trade of the district into their hands, and Mr. Clarke found out, when it was too late, that he had been duped.
The difference in this last example, clearly, is that the image is of a two-wheeled cart with an extra (third) wheel, rather than of a four-wheeled coach or carriage with an extra (fifth) wheel.
Ngram evidence of recent changes in idiomatic usage
An Ngram chart of "like a fifth wheel" (blue line) versus "like a third wheel" (red line) for the period 1900–2005 looks like this:
As the chart indicates, the ascent of "like a third wheel" since 1970 (and especially since the middle 1980s) has been quite dramatic. I suspect that the change is due to increasing use of the phrase in situations involving a (presumably romantic) couple plus a third person; under those particular circumstances, equating the odd person out with "a fifth wheel" poses a numerical problem, given that only three "wheels" are present.
Conclusions
The rapid rise of "like a third wheel" indicates that this version of the idiom has gained considerable momentum in popular usage since 1970. Still, usage of "like a fifth wheel" remains quite popular, too, so if we are in a transitional phase from having "like a fifth wheel" as the dominant form to having "like a third wheel" as the dominant form, we are only now at the crossover point in popularity between the two (in published writing, anyway).
So, the answer to the poster's question, "If you are the "extra" person in a situation, are you the third wheel or the fifth wheel?" appears to be that you are the third wheel if the reference vehicle is normally two-wheeled, and you are the fifth wheel if the reference vehicle is normally four-wheeled. At the moment, with the blessing of popular usage, you may invoke either sort of vehicle, and therefore you may be justifiably claim to be either the third wheel or the fifth wheel on it.
Solution 2:
As an idiom, the extra wheel is contextual to the count. If there are two other people, as in you are at dinner with a couple who are dating, you are the third wheel. If there are four other people, as in dinner with two couples on a double date or out with someone else's family of four, you could be the fifth wheel.
In general, vehicles with an even number of wheels (bicycles, motorcycles for two, cars, trucks etc. for four) are far more common than vehicles with an odd number of wheels (unicycle, tricycle/trike, and there's not common name for a five-wheeled vehicle), so both forms have some currency: Google Ngrams confirms that third- and fifth- are at least common. Wikipedia refers to third- as a corruption of fifth-, but doesn't provide a source.