To add in, pinging random website A and B gives nothing valuable to compare to pinging FFXIV's servers. Especially if they aren't even hosted in the same areas.

Packet loss can occur anywhere on the path from the OP's computer to the datacenter hosting FFXIV. I would recommend running a traceroute and if you haven't, post it in the Support forums thread. (Maybe here if you want other users with tech knowledge to take a look at where your connection is going bad.)

It may also be wise to try a trial of WTFast or another gaming VPN to see if it's possibly the route you get from your ISP to the game which is causing the packet loss.