Did you:
Try a Shift-Reload in your Browser?
Empty the Broeser's Cache?
Try to load the Stylesheet directly in you browser?
View the source of the page received?
I'm asking this because stylesheets and HTML are (amongst others) my daily business and these are the steps that usually give me the expected result or an insight where the problem lies.
Shift-Reload? Never heard of that. What does that do? I tried it but couldn't see it do anything.
Empty the Browser's Cache? Is this the same as deleting temporary internet files? I tried that, yes.
Load the style sheet directly into my browser? Do you mean doing something like this in the address bar? http://127.0.0.1:81/perl/secure/training_site.css
I tried that; it only opens the style sheet inside of an editor.
View the source page received? Yes, and the HTML code looks fine. In fact copying and pasting the resulting HTM code as it's own HTM file does correctly display the styles.
Despite the fact it didn't help, just a short explanation:
1. Shift-Reload (should) tell the browser not to use the cache.
2. No. Because there might be an In-Memory cache too. But I see: You're using IE!? I have no experience with that. Sorry.
3. Exactly. This is to verify you receive the correct CSS.
4. If this happens when you open the pasted code, then nothing can be wrong - to my understanding - with your code. It must be a caching problem because the only difference is the source where you get your page from.
Try to sniff on the traffic that's going on between your server and your browser and see what they are "talking" about.
But at least try to access your Server from another computer that never visited those URLs...