Let $T$ be a compact oriented torus, let $p\in T$ be a point, and let $T^*$ be $T - \{p\}$.

In Farb-Margalit's *Primer on mapping class groups*, in the discussion after Proposition 1.5 they say that "the homotopy classes of simple closed curves in $T^*$ are in bijective correspondence with those in $T$", but they give no hint of a proof.

I presume they left out the word "essential"? Even if we're talking about essential simple closed curves, what is this bijection? The only natural thing is to look at the map induced by the inclusion $T^*\hookrightarrow T$, but then this would mean that in each coset of the commutator subgroup of $\pi_1(T^*,q)$ (for some base point $q\in T^*$), there is a distinguished class which can be represented by an essential simple closed curve?

My questions are: What is the right statement (and how do you prove it)? Also, given an element $w\in\pi_1(T^*,q)$ written as a word in $x,y$ for some nice choice of basis $x,y\in\pi_1(T^*,q)$, is it possible to recognize when $w$ has a representative which is a simple closed curve? an essential simple closed curve?

I'm a novice in this area, so references would be appreciated as well!