The chai tea latte at Starbucks is made from the Tazo Chai Tea Latte concentrate. This product is available at supermarkets and online, and is mixed with milk to make a product not unlike what you have at Starbucks (minus the foam, as you pointed out).
So, what are these flavors (say, if you wanted to make this from scratch)? The website mentions that the black tea is malty. You could likely reproduce that with a maltier tea like Assam, though the box also says a "a rich blend of flavorful teas", meaning that a stronger blend like English Breakfast might be a closer match. It looks like you are on the right track with the spices as well: cinnamon, ginger, cardamom, and cloves are all listed, though the website mentions black pepper, star anise, and vanilla in addition, which ought to lift your cup of chai a bit as well.