"exhibition" vs. "exposition" vs. "exhibit" in AmEng
Solution 1:
In American English, originally the word "exhibit" as a noun was not applied to "goods or works of art" generally.
Instead, quoting Noah Webster's 1839 An American dictionary of the English language:
EX-HIB'IT, n 1. Any paper produced or presented to a court or to auditors, referees or arbitrators as a voucher or in proof of facts; a voucher or document produced. -2 In chancery a deed or writing produced in court, sworn by a witness, and a certificate of the oath indorsed on by the examiner or commissioner.
The 1904 Funk and Wangnalls A Standard Dictionary of the English Language does have a noun-meaning of "exhibit" of, among other meanings:
Any object or collection of objects placed on show; as, an agricultural exhibit
The current Webster's definition of EXHIBIT as a noun is:
1 : a document or material object produced and identified in court or before an examiner for use as evidence 2 : something exhibited 3 : an act or instance of exhibiting : EXHIBITION
So in the mid 1800's exhibition and exposition were interchangeable, and now all 3 are interchangeable.
Solution 2:
I will answer as an American layperson, from what I consider the most popular to the least popular in common use.
Exhibition
A public showing of a collection of artifacts, often in a museum or gallery. Less commonly a performance on stage, in which case typically phrased as e.g. "dance exhibition" rather than simply "an exhibition" (which absent other context seems to imply artwork).
Exhibit
A single artifact being shown, or a single collection of artifacts within a larger set (or exhibition), as in "Let's visit the dinosaur exhibit first."
Exposition
Not very common in its full form, but somewhat popular as "expo" as "the car expo" would be a showing of motor vehicles for sale. Saying "exposition" when talking about an "expo" would likely sound unnatural in colloquial American English. Used in its full form within the art world to indicate a large display of artifacts with a single theme, as in "An exposition of young American photographers." May refer to something considered novel, rather than the well-known: "An exposition of Warhol's works" sounds stilted if the audience is understood to have seen the works before.
Solution 3:
Exhibition vs Exposition: http://www.ziggurat.es/cat/lecciones_ingles/index.asp?id=727
"exposition" is a title given to some exhibitions, usually showcasing modern technology. You should never refer to an exhibition as an exposition, unless that exhibition is specifically titled as an exposition. The two words are not interchangeable.
Exhibit vs Exhibition: http://grammarist.com/usage/exhibit-exhibition/
Basically, an exhibit is a part of an exhibition, and, conversely, an exhibition is a collection of exhibits.