$x \in R[x]$, $x^2 \in R[x]$, since it is a field any operation * b/w these two's results must also be in $R[x]$, but $x^-1$ is not in $R[x]$
So clearly polynomials do not form a field.
However, Professor Ghrist says that it is a polynomial field at 1:04 of this video https://www.youtube.com/watch?v=pePJKl6EFDo&t=63s
Am I missing something?
That's a really bad expression since it tempts the listener to confuse two totally different issues. What he mean by "polynomial field" has nothing to do with "fields" in algebra. Here he draws an analogy to vector fields: https://en.wikipedia.org/wiki/Vector_field
A vector field assigns to a point a vector, in yours/his case the "polynomial field" assigns to a point a polynomial: the Taylor expansion around the point.
That's why he calls it a "field". But this is not standard usage of language, no algebraist would use this "terminology" here, but the informal analogy is ok.