"Half-Life: Caged", "Half-Life: Restored", "Half-Life: Before", "Half-Life - The Freeman Chronicles"

Mods, and a fan video (there was a time were steam were seeming to look into hosting movies videos, you can find some of that on the store as a legacy stuff i guess).

"Half-Life 2: Downfall", and "Half-Life 2: Update" Mods

There is only the orginal half life series

Which started with the Half life, then come the stand alone expansions Blue Shift and Opposing Force and then Half life 2 with it's stand alone expansions Episode one and two.

You got a remake of Half life called Black Mesa not made by valve.

You can play Half life 2 without knowing the first one, although some refs will go missed, but nothing of grand importance.