Where I'm from, that "green turn arrow" is called a "protected left". While I can't say I've ever heard unprotected left, the term makes sense as a logical extension of the lexicon. (And as a Bay Area resident… I chuckled at the remark about being able to make unprotected lefts.)

I will occasionally abuse the term with the term "protected right", to mean waiting at a red light where the cross traffic moving leftwards has a protected left. (because the cross traffic's left is protected, the only conflict with a right-turn-on-red is someone in the cross traffic U-turning.)