Don't rely on a dictionary for colloquial (or natural) language to be an accurate source of mathematical terminology. There are plenty of times that a mathematical term has a technical meaning which is not what you might expect based on the "usual" interpretation of the word or its parts. In the case of "polynomial" this is a simple instance of weakening, where "poly"is interpreted as "possibly many" rather than "many" (similarly to the use of "or" to mean inclusive "or" in logic); here are many more drastic examples of this, where the connection between term and meaning is tenuous or even nonexistent, my personal favorite being the use of the word "weasel" in set theory.
Wikipedia's definition of polynomial is in fact correct. (That's not to say that wikipedia is infallible, but it's pretty good generally for mathematics; one should really confirm wikipedia's definition by looking at an appropriate text.)
Incidentally, just to drive the point home: weasels are so called because they are like mice, but larger. Why are mice called that? To quote a famous philosopher, "I'll tell you ... I don't know."
(There are two competing stories that I've heard. One is that the mathematician in question, Jensen, made a typo and "we call such sets nice" turned into "we call such sets mice," and he liked it so much he kept it; the other is that Jensen simply wanted a word that had no prior meaning in mathematics and picked it mostly at random. My understanding is that Jensen himself has told both stories at different times.)