In Steenrod's The Topology of Fibre Bundles, there are two different notions of covering homotopy. One is furnished in Theorem 11.3:
Let $\mathcal B,\mathcal B'$ be two bundles having the same fibre and group. Let the base space $X$ of $\mathcal B$ be normal, locally-compact and such that any covering of $X$ by open sets is reducible to a countable covering. Let $h_0\colon\mathcal B\to\mathcal B'$ be a bundle map, and let $\bar h\colon X\times I\to X'$ be a homotopy of the induced map $\bar h_0\colon X\to X'$. Then there is a homotopy $h\colon\mathcal B\times I\to\mathcal B'$ of whose induced homotopy is $\bar h$, and $h$ is stationary with $\bar h$.
The other is furnished by Theorem 11.7:
Let $\mathcal B'$ be a bundle over $X'$. Let $X$ be a $C_\sigma$-space (i.e., satisfying conditions specified in Theorem 11.3), let $f_0\colon X\to B'$ be a map (where $B'$ is the total space of $\mathcal B'$), and let $\bar f\colon X\times I\to X'$ be a homotopy of $p'f_0=\bar f_0$. Then there is a homotopy $f\colon X\times I\to B'$ of $f_0$ covering $\bar f$ (i.e. $p'f=\bar f$), and $f$ is stationary with $\bar f$.
The second one is a corollary of the first one. However, as far as I've searched, the second one is so-called homotopy lifting property, and is used to define fibrations. I wonder whether the second one (in some sense, say with some mild extra conditions) implies the first one, and the value of the first definition in modern literature (since Steenrod is somewhat archaic).
Any help? Thanks!