I took a look at your site. The problem definitely seems to be with thumbnail creation. The case isn't solved, but I do have a some suspects
In phpinfo(), under the Core section, you'll see there's a very long list of disable_functions. These are all php functions that your server won't allow. Probably for security reasons, though most hosts don't disable anything. Amongst the looong list you'll find these 4: system, shell_exec, exec, passthru. phpThumb will use any of these 4 to run the ImageMagick 'convert' command, but in this case it doesn't have any available way to run the command. So even though ImageMagick is installed, phpThumb can't use it.
This is where things get strange though. phpthumbof (a wrapper for phpThumb) works fine! I set up a test page so you can see exactly what it's doing.
But when the same phpThumb is called via a connector (i.e. thumbnails in the manager), you get a 500 error almost all of the time. Though, a couple times I actually saw a thumbnail pop up in the file browser (?!?). Also core/cache/phpthumb/ does have images in it, though it's possible you uploaded these when you transferred the site from your local environment to the server; I don't have FTP access and can't check the file times to make sure.
I have no idea why that is. Possibly the server error log—different from the MODX Error Log—has some more clues. My guess is that it's caused by some (security) setting on the server. As a side note, it appears this server may be running modSecurity: I noticed I was unable to save a resource when it had particular content in it.
Also several times I had this odd thing happen: I'd successfully save something, go back and look at it, and see the old version. Disable the browser cache, navigate somewhere else, come back, still the old version. Then logout or switch browsers, login, check the file again, and this time it's the new version. Strange...
I'll stop rambling here, but the short of it is, this host seems to have some odd settings, likely the cause of the thumbnails problem. If they have good tech support you might be able to figure it out together with a sysadmin. However, I'd seriously think about switching hosts if that's a possibility; you'll save yourself a lot of frustration!