No need to confuse yourself. Transocean and Deutag are DRILLING CONTRACTORS. They own rigs and do the drilling. Halliburton and Schlumberger are Oilfield service companies. They do not own rigs. They simply supply products that help the operator(with the aid of the drilling contractor) get to the pay-zone. Halliburton and Schlum provide things such as directional drilling tools and services for horizontal/s shaped wells, completion tools like frac tools, wireline services for gyros/surveys etc.