How to enter SWTOR's Flashpoint "The Esseles" in Veteran mode?

The missions you currently have in your log are both for Veteran Esseles. Or rather, the mission "[FLASHPOINT] The Esseles" is difficulty-agnostic - it is the same on all difficulty levels. When entering a flashpoint on Story mode, you should have a "[STORY] Flashpoint:..." mission and (in most cases) will receive server messages about the GSI Combat Support Droid. (I've actually not done Story mode flashpoints in a group before, so I don't know if the droid is only for true solo runs.)

Note that the "two players recommended" in the Veteran Esseles description is not an error; unlike every other flashpoint, The Esseles (and its Imperial counterpart, The Black Talon) was originally designed for two players.