$\delta$ hyperbolic geodesic metric spaces
You're right to suspect that definition for $\delta$-hyperbolicity: it is the wrong definition.
Given the geodesic triangle with sides $a,b,c$, the correct definition is not a bound on $d(a, b \cup c)$, instead:
For each $y \in a$, $d(y,b \cup c) = \inf\{d(y,x) \mid x \in b \cup c\} \le \delta$.
One could, if one so desired, write this as a "sup-inf":
$\sup_{y \in a} \bigl( \inf\{d(y,x) \mid x \in b \cup c\} \bigr) \le \delta$
(In fact this is even a "max-min", using compactness of the sides of the triangle.)