The genus referred to in the above paper by Baldwin and Levine (and in many related papers in knot Floer homology) is the Seifert genus, also known as the three-genus $g(K)$. This is the minimal genus over all Seifert surfaces bounded by the knot.

The statement that the genus of a thin knot is the degree of its Alexander polynomial is a consequence of two major theorems, both due to Ozsváth and Szabó. The first [1] is that knot Floer homology categorifies the Alexander polynomial and and the second [2] is that it detects the (Seifert) genus of a knot, or more generally the Thurston norm of a three-manifold. In particular, the Seifert genus $g(K)$ is the largest grading $s$ (an integer) where the knot Floer group $\widehat{HFK}_n(K, s) \neq 0$. For homologically thin knots, this integer is also the maximum degree of the symmetrized Alexander polynomial.

Note that the Seifert genus $g(K)$ is not quite, as you said, "the minimal genus taken over all spanning surfaces of the knot" because spanning surfaces may be orientable or non-orientable, whereas Seifert surfaces are necessarily orientable. (The `genus' of a non-orientable surface would be the the minimal number of projective planes required.) All the above statements pertain to orientable surfaces. The smooth four-ball genus and smooth four-dimensional crosscap number also shows in up Heegaard Floer homology too, though I think people are pretty explicit about those being smooth, four-dimensional and/or unoriented.

[1] *Ozsváth, Peter; Szabó, Zoltán*, **Holomorphic disks and knot invariants**, Adv. Math. 186, No. 1, 58-116 (2004). ZBL1062.57019.

[2] *Ozsváth, Peter; Szabó, Zoltán*, **Holomorphic disks and genus bounds**, Geom. Topol. 8, 311-334 (2004). ZBL1056.57020.