It's indeed true that the action by lifting loops and the action by deck transformations agree exactly when the fundamental group is abelian. This statement is a bit vague, so let me be precise.
Let $X$ be a space with universal cover $Y\stackrel{p}{\longrightarrow} X$, and choose basepoints $x_0 \in X$, $y_0 \in Y$. Then we can identify the fiber $p^{-1} (x_0)$ with $\pi_1 (X, x_0)$ using path lifting: a loop $\gamma$ at $x_0$ lifts to a path $\tilde{\gamma}$ starting at $y_0$, and we identify $[\gamma]$ with $\tilde{\gamma} (0)$. Now the action of $\pi_1 (X, x_0)$ on the fiber, via lifting, corresponds to right multiplication in $\pi_1 (X, x_0)$. On the other hand, the group of deck transformations of $Y$ is also isomorphic to $\pi_1 (X, x_0)$, by sending a loop $\gamma$ to the deck transformation taking $y_0$ to $\tilde{\gamma} (0)$. Now under the identifications of both the deck transformations and the fiber with $\pi_1 (X, x_0)$, the action of deck transformations on the fiber corresponds to left multiplication in $\pi_1 (X, x_0)$.
Checking these statements is a worthwile exercise. In some sense, this is easier to think about if you initially just think of actions as functions that assign group elements to bijections, and then you can later worry about left versus right.
In any event, a group G is abelian if and only if for every element $g\in G$, left multiplication by g and right multiplication by g are the same function $G\to G$.
I'll also note that any left action of a group $G$ on a set $S$ can be converted into a right action by setting $s\cdot g = g^{-1} \cdot s$, but the above discussion shows that if you convert the left action of $\pi_1 (X, x_0)$ on the fiber (via deck transformations) into a right action, you definitely do not get the lifting action.