You are mostly right, but there are different types of stainless steel. To be honest I did think they all had aluminum but its ferritic stainless that contains aluminum.
Copied from wiki:
Ferritic stainless steels generally have better engineering properties than austenitic grades, but have reduced corrosion resistance, due to the lower Chromium and nickel content. They are also usually less expensive. They contain between 10.5% and 27% chromium and very little nickel, if any, but some types can contain lead. Most compositions include molybdenum; some, aluminium or titanium. Common ferritic grades include 18Cr-2Mo, 26Cr-1Mo, 29Cr-4Mo, and 29Cr-4Mo-2Ni. These alloys can be degraded by the presence of σ chromium, an intermetallic phase which can precipitate upon welding.
The type of stainless most commonly used for cookware is austenitic stainless, again copied from wiki:
Austenitic, or 300 series, stainless steels make up over 70% of total stainless steel production. They contain a maximum of 0.15% carbon, a minimum of 16% chromium and sufficient nickel and/or manganese to retain an austenitic structure at all temperatures from the cryogenic region to the melting point of the alloy. A typical composition of 18% chromium and 10% nickel, commonly known as 18/10 stainless, is often used in flatware. Similarly, 18/0 and 18/8 are also available. Superaustenitic stainless steels, such as alloy AL-6XN and 254SMO, exhibit great resistance to chloride pitting and crevice corrosion due to high molybdenum content (>6%) and nitrogen additions, and the higher nickel content ensures better resistance to stress-corrosion cracking versus the 300 series. The higher alloy content of superaustenitic steels makes them more expensive. Other steels can offer similar performance at lower cost and are preferred in certain applications.