Why a small two-ship action still counts as a battle

Is a two-versus-two ship engagement really a battle rather than a skirmish?

Yes. In the usage explained here, an action fought through to a decisive end is a battle regardless of how few ships take part, while a skirmish ends with both sides withdrawing.

Fleet size does not determine whether an action deserves the label of battle. Phoebe and Cherub fought until Essex surrendered, so the encounter reached a decisive conclusion rather than ending as a probing clash with both forces withdrawing. A skirmish, by this definition, ends with both sides withdrawing. Small Age of Sail actions do not need large formations to count as battles.