US guides
The report is how you know which diamond is coming to your US address. Grades without a report number are not a purchase.
US shoppers usually recognize GIA, especially on natural diamonds. IGI is common on lab-grown loose stones. Either report is usable if it is the report for the stone on the page and you can open it.
Compare the report number on the listing with the inscription the lab describes. If you are asking a US jeweler to set the stone, they will look for that same number.
Carat, color, clarity, cut, and measurements should be the ones you compared. Then request that diamond so the US shipment is tied to that certificate.