the PDF version comes through with the black dog-tag shape and black text. The svg version has the black dog-tag shape and a grey shape for the interior of the tag. You will want to add a new shape (or two) in a different color where you want the laser to cut. It is fairly straightforward to do in vector-editing programs like Illustrator, inkscape, corel, etc. Do you have access to any of those? if not, what vector software do you currently use?
In illustrator, starting with the SVG version that you posted, I made a duplicate of the black layer, then turned off the grey layer and one of the black layers. Using the Direct Selection tool, I clicked one of the nodes in the inner cutout and then hit delete twice, so that I was left with a mostly solid black oval and the hanging hole. I set the fill to None and the stroke to red. Then I turned the other two layers back on.
Feel free to use this as a starting point:
from there, I might make the hanging hole a unique color so that I could set it to cut before the outline. I might ditch the grey layer altogether and just add new text, or set it to score, or punch the text out of it if I wanted the letters to stand proud.