How do I interpret the second item in the ingredients list in this recipe?  1 cup Grape Tomatoes, Halved Do I measure out 1 cup of grape tomatoes and then cut each one in half? Or should I start cutting them in half and put all the half pieces into my measuring cup until it's full? I can certainly fit more grape tomatoes into my measuring cup if they are all chopped in half than if they are whole. I realize of course that most of the time the difference is probably too small to matter. But what if it did matter? 
It probably doesn't matter.  I bet the difference isn't more than a few grape tomatoes. The way I read it, if the author wanted you to cut first, it would state "1 cup halved grape tomatoes," indicating that you should measure the halved tomatoes. When it is important, a good recipe would specify in the procedures section.  If accuracy of this sort is necessary, a good recipe would be even more useful if it listed ingredient weights, rather than volumes. 