The error message you are getting about not being able to write has nothing to do with your WordPress installation. That error is coming from the main website. So the host’s answer doesn’t apply here at all.
1) the name of the cache folder is just cache correct, no period in front of it?
2) another possibility was hinted at in the article I linked you to – the ownership of the cache folder that you created. If the website was created by someone else you may not have the necessary permission to write the cache folder in a way that the code running on the server can access. You can tell the webhost the issue is simplepie running on the website can not write its cache files ON THE WEBSITE in the cache folder. It has nothing to do with WordPress. You could be pulling in RSS from CNN and the same problem would occur because the problem is with the cache folder ON YOUR SERVER not the one the WordPress installation is on.
From the article:
A note about caching
SimplePie provides a great caching mechanism so that feeds don’t have to be retrieved on every page load. This requires a writable directory on the server where the feeds can be stored. By default this location is ./cache, meaning a directory called “cache” in the same directory as the file calling simplepie.inc. Be sure your web server can write to this directory, either by changing the permissions, ownership, or some other means, as appropriate for your setup
If changing the permissions to 777 didn’t do it, write the host again and explain this to them.