Yes at 13 you're considered to be a teenager.
When you hear the word teen in the age then you're a teenager so, 13,14,15,16,17,18 and 19 are considered to be the teenage years although by age 18 you're considered to be an adult but you don't reach full adulthood until age 21.
At age 12 you're considered a tween by when you turn 13 that's when you are now a teenager.
You stay a teenager until you're 20 even though legally you're an adult at age 18.