A condition for a covering map to be regular

Question

We said that a path-connected covering map $p:E \rightarrow X$ is regular if:

$\forall e \in p^{-1}(x_0): p_{\sharp} \pi_1(E,e)$ is a norm subgroup of $\pi_1(X,x_0)$.

or equivalently: If closed paths in $X$ lift to closed paths in $E$.

I highly doubt that this second condition is equivalent to the first one. The reason is that universal covering maps should fulfill the first condition, but they only fulfill the second condition if the path we started with was nullhomotopic. Thus, I would be interested in your opinions and whether you think that one can fix the second condition, if it is wrong.

Answer 1

The statement is wrong, of course. The correct one is that a covering is regular if and only if for every loop in the base one lift is closed iff all lifts are closed. (One still has to define properly what a closed lift means.)