The second time that the 2nd Doctor used a sonic screwdriver on-screen was in the story “The War Games.” In the video below, the Doctor is trying to prove that he is from another planet by demonstrating his “alien” technology …. his sonic screwdriver. Notice how simplistic it looks. Most likely it was just a pen light that was very common at the time of filming.
Unofficially in the in Doctor Who fandom the various varieties of sonic screwdrivers are referred to by their mark or model number. Therefore, the first sonic screwdriver used by the 2nd Doctor is known as the Mark I and the most recent used by the 11th Doctor is referred to as the Mark VII.
The use of the term mark in this case is used to mean model number or model of a product of manufacture, such as a weapon or car. Which seems appropriate as the sonic screwdriver is not only a tool but sometimes referred to as a defensive weapon of the Doctor’s.