...although it gives the popup it always shows an empty cart and distorted looking popup.
That sounds like a plugin or something else in your WP setup is using its own "lightbox" overlay to display the link's target URL within an iframe, rather than calling our own cart-overlay script.
What I'd really like is the little shopping cart that displays top-right in the default popup. But a link will do. Surely this has come up before? It's hard to think of something more basic in e-commerce. It's also part of your shop demo...
That comes down to the distinction between a shopping cart vs. a storefront. For the FatFreeShop hosted storefronts in our site's Marketplace, we control the entire page layout and thus, as part of that layout, we can generate a cart icon with quantity, as you saw. However, when you're just using our Cart buttons within a site built independently of us, we can only control how the cart overlay appears when a button is clicked, so adding extra things to your own sales page(s) would be up to you.
That said, there is a way to display the cart quantity wherever you wish in your page, which can also be made into a clickable View Cart link, though any icon and/or text in that link would be up to you: