Q&A: Searching for a 2 Ct. Round Brilliant with Around $13k Budget

Hi, I have read numerous articles on the site and your advice has been incredibly helpful. I am looking for a round cut diamond with the following spces:
-2ct
-SI1 (or better)
-J color (or better)
-VG/Ideal cut

I would like to spend around $13K. I found several options mostly searching Blue Nile and wanted to get your opinion on the stones and any advice on my search criteria. The setting is white gold with a small stones on either side of the band. The diamond links are pasted below.
Blue Nile LD03622083
Blue Nile LD03716793
Blue Nile LD03726231
Blue Nile LD03706721

Thanks so much for the help! 

Thank you for the response and input into the diamonds I found on bluenile. I read the suggested article (and just about every other one on the site) and moved my search to James Allen. I found several other selections I would like your opinion on:
1. https://www.jamesallen.com/loose-diamonds/round-cut/2.02-carat-j-color-si1-clarity-sku-240420?a_aid=dmnd1357
($13.8K)

2. https://www.jamesallen.com/loose-diamonds/round-cut/2.01-carat-j-color-si1-clarity-excellent-cut-sku-261023?a_aid=dmnd1357
($14.5K)

3. https://www.jamesallen.com/loose-diamonds/round-cut/2.01-carat-j-color-vs2-clarity-very-good-cut-sku-201330?a_aid=dmnd1357

4. https://www.jamesallen.com/loose-diamonds/round-cut/2.00-carat-j-color-si1-clarity-excellent-cut-sku-263166?a_aid=dmnd1357

I called JA about the first two and they confirmed they are completely eye clean. What do you think about these 4? Are SI2 worth looking at or stay in the SI1 or better range?

Thanks again. 

Thank you!