Mike,
Very, very sorry for not getting back sooner. A new release (mainly for Joomla 1.5 support, but with some fixes for Joomla 2.5 as well) and holidays have slowed us down a little.
Page is the right way to go.. post
Going to try to take your posts in order, even though it seems like you changed your mind along the way

A Facebook Page is a great thing to have. They are mostly anonymous in that user's won't be able to see the user that created the Page in any obvious way. I say 'mostly' because I can't guarantee that there isn't some way they could figure it out, but it's certainly not 'obvious'. The benefit of a Page is that when you post a status update, that message is likely to go to many of the users that have 'Liked' the Page. That's great from a promotion standpoint and well worth having, even if it's not the 'Like' button you use on your site (as you sound like you're going for in later posts).
Having a Page is generally a good idea in general, even if it's not something you actively promote. People will find it, Like it, and it will generally grow, organically, over time.
Let's start over... post
The App really is completely disconnected from any Like buttons you see. So, don't even think about the App going forward. The Like button can be used to 'Like' any URL on the web. So, you can have a Like button that is for your FB Page, your home page, or any article or sub-page on your site. A Like button for your homepage is great because, when it's "Liked" by a user, it can show up in their (and their friends) feed. Great, great promotion with title, description, image, etc of your site. Sub-Page Like's are great as they can do the same to get interest in very specific content on your site. A Like for your Page is great because (as above) status updates from your Page can get pushed to your users/friends feed.
So, there are advantages, basically, to having Like buttons for lots of different pages. However, starting off with a Like button to just your home page is a good ole' idea. To do that, you can create a JFBCLike module or use the JFBCLike tag on any page of your site and simply set the URL for it to reference to be your home page.
So tell me what you want, what you really really want.. post
I would recommend 2 things:
#1) Create a FB Page: Use the JFBCFan module that references that FB Page on your website. That way, when you update your FB Page, that message will get pushed out to your users.
#2) Much lower priority, but still good... create a Like button that references your home page. Put it in the footer or something like that. As it gets clicked, your like count for your home page will grow. As it increases, you can improve it's position to somewhere higher as a badge of 'look at our like count' when new users come. Additionally, search engines are starting to look at like counts more for rankings, so getting some here is good. You won't be able to push content to these users, but it's a good sign of overall popularity.
Hope that all helps explain things, but again, as always, let us know if you have more questions or need more info.
Thanks,
Alex