What's up with this site and not updating the character thumbnail?

The reason that the images aren't resolved quickly as they are on Lodestone is because of how the images are displayed on Lodestone opposed to the forums here. Lodestone forces a server cache reset based on a method that involves passing a random value (usually a timestamp) at the end of the server request to update the cache.

If you take a look at Lodestone and you copy your image URL and compare it with the one on the site here, you'll notice two differences:
1. The resolution is 50x50 on Lodestone and 96x96 here
2. The addition of the timestamp for the cache update.

Your normal forum URL for your image is: http://img2.finalfantasyxiv.com/f/93...0fc0_96x96.jpg
Your Lodestone image URL is:
http://img2.finalfantasyxiv.com/f/93...jpg?1409991393

You'll notice the timestamp appended to the end, but this ISN'T required for force the cache update, you can do anything to force it, it's just matching a URL pattern, that includes just typing a ? at the end of the URL.
http://img2.finalfantasyxiv.com/f/93...fc0_96x96.jpg?

As for fixing your problem... well... there's not much you can do about it I'm afraid. The reason that the servers cache the images on the forums for much longer is likely because the forums are a high traffic area where your image is more likely to be seen, and so to save bandwidth costs the cache is kept for longer than on lodestone where they force an update to keep Lodestone as current as possible.

tl;dr - Server cache problems, you'll have to wait longer... sorry.