Exergonic vs Endergonic Reactions
The reaction to consider is the transformation of chemical energy into visible radiation (light) and a little of heat.  A glowing stick consists of at least two compartments which -- on purchase -- a supposed to be kept separate.  The snap only breaks the wall between the two so that chemicals may mix with each other to start chemoluminescence (work principle), a principle similar to a firefly in late summer, too.
To decide if the reaction is exothermic, endothermic; exergonic or endergonic you have to compare the energy of initial state (when the chemicals are separated, no emission of light) with the one after the reaction.  There may be an energetic barrier to trigger the reaction, equally known as activation energy, but in its current form, this detail is not relevant to answer the question.
Take an analogy employing potential energy you might recall from the physics lecture: imagine you ate in a restaurant on a mountain and now walk back home.  It is not relevant that some energy was required to get up from the table, it is only about the comparison between
$$\mbox{state A «being in the restaurant»} \ce{-&gt;} \mbox{state B «back in the valley» } + \mbox{energy}$$
Even the pathway, straight downhill, or passing some smaller hills to reach the destination is not important here.  The naming then is, by convention, a formalism.