Let (i)$\left(F \xrightarrow{\quad \quad} E \xrightarrow{\quad \pi \quad}B \right)$ be any fiber bundle.
Wikipedia has a different parlance when defining some fiber bundles. Take this excerpt from the article on bundle maps:
Let (ii)$\left( E \xrightarrow{\quad \pi_E \quad } \mathfrak M \xleftarrow{\quad \pi_F \quad } F\right)$ be fiber bundles over a space $ \mathfrak M$. Then a bundle map from $E$ to $F$ over $\mathfrak M$ is a continuous map $\left( E \xrightarrow{\quad \varphi \quad} F \right)$ such that $\bigg( \pi_F \circ \varphi = \pi_E \bigg)$.
(a)
In lieu of (i): "$\left( E \xrightarrow{\quad \pi_E \quad} \mathfrak M\right)$ is a fiber bundle over a space $\mathfrak M$" is only a terse way to state that we have a bundle $\left(F \xrightarrow{\quad \quad} E \xrightarrow{\quad \pi \quad} \mathfrak M \right)$? If so, then what is the $F$ that Wikipedia is using? Like $E$ another total space?
(b)
Assuming (perhaps incorrectly) that $E$ and $F$ in (ii) the same $E$ and $F$ in (i), then is $\varphi$ a map from the fiber to the total space?
I saw no clarification for any of these questions on the main Wikipedia page for fiber bundles. Thanks in advance!