Quick solution is to choose a score, and then go into manual settings and adjust the focus on that one line to 1/4-1/2" above or below your design. A defocused score will be shallow and wide.
The actual solution to the engrave question is that you need to make the green line into a wide path, rather than a wide stroke. In Inkscape it’s under the Path menu and it’s called Stroke to Path. It should be something similar in Illustrator. It’ll make into an object with two edges that the machine can then engrave between.
Ah hah, I did a quick search and it’s called “Outline Stroke”