The knowledgebase article is pretty straight forward.
Under the module manager in the admin click - Catalogs, and then you can see the "Product Search Form" to input the form. You can grab that html and do with it and place it where you need to.
Just like a site search you can send it do a new page for the results etc. All of that is in that knowledgebase article.
What do you feel is missing or not explained?
You're right. That article is broken. I'll note that down and get someone to fix it.
Anyway, there are the steps:
Go to a web page or the template you want to have your searh form on and click on Add module to a web page/template. You can also click on the little blue circle with the + sign it. This will open the module manager in the seperate frame on the right.
You then click on Catalogue > Product Search Form > Insert. This will insert the search form into your page/ template wherever you've possioned your cursor.
You can then style your form as per the article you've found.
BTW (by the way), this is how you insert all modules in BC, through a module manager.