I don't think it's incorrect per se to uses spicy and hot interchangeably. It's just maybe not common to all dialects of English. Where I'm from nobody would refer to a dish flavored with your first list as spicy.
No cinnamon, cardamon, cloves, nutmeg, star anise are ingredients common to every spice rack and every grocery store's spice aisle in the English speaking world. Even one of Starbucks most popular drinks is a "Spiced Pumpkin Latte" which consists of pumpkin, cinnamon, nutmeg and clove, served over ice. During the European Age of Discovery the "Spice Island" i.e the Molucas in what is now Indonesia were so important because they were the source of nutmeg and cloves. Spice is added to food for either flavor or aroma. Heat is a sensation, its felt. Lastly Capsaicin - the compound in the ribs and oil of peppers doesn't add flavor just a sensation. There is no such thing as a salty, sweet or bitter heat. And it's not just English, in Spanish something that had heat such as a hot salsa would be considered "piquante" but not spicy.
