If the dancers do the right thing (and have fun) when I say the words then I'm calling it correctly.

If the dancers are confused or not having fun, then I am not calling it correctly.

If I can use fewer words to call it correctly then I try to do so.

Sometimes my words are directed to the active couples, sometimes to the inactives, sometimes to the gents, sometimes to the ladies, and of course sometimes they apply to everybody.

Always they are intended to help the dancers enjoy themselves.

Getting back to the original question.

If I were to call "pick up your neighbor" and I observed the people standing still somehow trying to "pick up" the person passing by in the allemande, then I would change the words I use.  Until then, I use the words that I "always" use and that the experienced dancers are used to hearing. 

And as for "courtesy turn", those two words are sufficient.  There is no need  (and usually no time) to mention that one dancer is the turner and the other the turnee.

Dale