On page 59 in Lee's "An Introduction to Smooth Manifolds" the author writes,
"Let $E$ be a smooth vector bundle over a smooth manifold $M$, with projection $\pi:E\to M$."
I thought the vector bundle in this case would be $\left(E,M,\pi\right)$, making $E$ the total space rather than the vector bundle, or am I wrong?
You're right; this is just a common abuse of terminology. This is similar to how we frequently talk about a "group $G$" (when the group is really the pair $(G,\cdot)$) or a "topological space $X$" (when the topological space is really the pair $(X,T)$, where $T$ is a topology on $X$).
(Actually, it is not even accurate to say that the vector bundle is $(E,M,\pi)$, since you also have to specify the vector space structure on each fiber.)