Is a twisted knee the same as a sprained knee?

Yes if.. If no tearing/rupturing of the ligament/tendon, then it is called sprained/strained knee. Strained is a bit more severe, but still no tearing. Ice, rest, gentle range of motion/stretching...If symptoms worsen, see your doc...
Close. A sprain describes stretching or tearing of ligaments; a knee sprain is a very general term, often used before a specific diagnosis is made (ie which specific ligament or ligaments are injured). Sprains can occur from twisting, but twisting can also result in other injuries that are not sprains, such as cartilage tears. So sprain is a type of injury, while twisting is a mechanism of injury.