Zack here, one of Shopify's employees! I changed where the "<input type="hidden" name="type" value="product">" tag was added (from Line 162 to Line 170), and that seems to have rectified things. The React Theme is a little bit different from the Theme that the guide describes, and I thought that adding the code just above the closing </form> tag ought to do the trick.
And it did! I added spaces before and after where I added the code in, so you can see it easily :)
All the Best,