If I understand what you want to do correctly, here is a simple, off-the-top of the head solution.
1. Using whatever method you found above to "eliminate" a category from your category list, wherever your category list is shown, if you don't want that category shown in that list.
2. Create a manual link to your chosen category (let's call it Alpha with an ID number of 5) in your WordPress Theme header template file where you want the link to the category to appear.
<a href="/index.php?cat=5">Alpha</a>
3. Using the Template Hierarchy, create a Category Template called category-5.php and put it in your WordPress Theme folder.
Make it "look" however you want it to look by copying the code and design elements from your index.php or category.php. Be sure to include the "includes" for the header, sidebar, footer, etc. You may want to change the_content() to the_excerpt() to show only excerpts not full posts. You can include custom text or whatever you want on that template file.
4. When a user clicks on the Alpha link, it will automatically call up the category with ID number five and show only the posts within that category. There is nothing you have to do to the WordPress Loop or anything.
If category 5 shows up in any category list, like in a category list on a post, any time anyone clicks the Alpha category, WordPress will automatically display that custom category page.
There is nothing more "special" you have to do.
Part of the confusion on this was your use of "page". In WordPress, a "Page" is a special type of web page and a "page" is another word for "web page", which is any web page on your blog. A "post" is a web page with your post content, and that uses the WordPress Loop to be generated on your WordPress blog web pages.
Putting a self-generating list of posts from a specific category on a Page makes things complicated. However, customizing a category page is easy, no messing with the Loop.
PART II
If you would like a list of the posts ONLY in the Alpha category to be seen in a special section on the front page of your blog, that's a different issue. There are a lot of ways you can do this. If you want it to appear on every page, see query_posts or check out Customizable Post Listings WordPress Plugin by Scott Reilly to show the posts from a specific category, from inside or outside the WordPress Loop.
If you want the list of Alpha's post to appear ONLY on the front page and not on the other web pages on your blog, then you can use a conditional tag statement in your sidebar, header, or wherever, that lists the last 5 posts in that category:
<?php
if (is_home()) { ; ?>
<h5><a href="/index.php?cat=5">Alpha</a></h5>
<ul>
<?php
$myposts = get_posts('numberposts=5&category=5');
foreach($myposts as $post) :
?>
<li><a href="<?php the_permalink(); ?>">
<?php the_title(); ?></a></li>
<?php endforeach; ?>
</ul>
<?php }; ?>
I shouldn't write code this late at night, so someone please fix this if I messed up.
There are a lot of ways you can do this, including with the Customizable Post Listing WordPress Plugin or using a feed link to that specific category in a Widget or manually generated so you wouldn't have to use all that code. Try one of the various Feed/Syndication WordPress Plugins to help you do this to make it easier.
Hopefully this will help you do what you want. You can also set up a whole separate blog within a blog, but you are only messing with a specific category that you want "within" your current blog. This also allows you to change your mind when you are done highlighting that specific category, giving you more flexibility.
Am I on the right track?
