Eccentricity, radius and diameter as defined in "Graph Theory and Complex Networks: An Introduction" (van Steen, 2010):
Consider a connected graph G and let d(u,v) denote the distance between vertices u and v. The eccentricity e(u) of a vertex u in G is defined as max{d(u,v)|v € V(G)}. The radius rad(G) is equal to min{e(u)|u € V(G)}. Finally, the diameter of G is the maximal shortest path between any two vertices: diam(G) = max{d(u,v)|u,v € V(G)}.
I'm using the Euro sign € where I mean "within set"
Does this mean that "radius of G" = "smallest eccentricity" and "diameter of G" = "largest eccentricity"?
Indeed they do. As is stated in the book, $\DeclareMathOperator{\rad}{rad}\rad(G)=\min{\{e(u)|u\in V(G)\}}$, so by definition, "radius of $G$" = "smallest eccentricity." As for the diameter,
$\DeclareMathOperator{\diam}{diam}\diam(G)=\max{\{d(u,v)|u,v\in V(G)\}}\\=\max\{\max{\{d(u,v)|u,v\in V(G)\}|u\in V(G)}\}\\=\max\{e(u)|u\in V(G)\},$
so "diameter of $G$" = "largest eccentricity."