I was induced at 37+4 weeks and it didn't progress so elected for a c-section (and have not once regretted my decision). I think something important to be aware of is the risks of attempting to deliver a large baby - shoulder dystocia and other complications can be VERY dangerous and it is not worth it to risk it. Second baby was delivered by c-section at 37+5 weeks. Both very healthy, both more than ready.
Full term is defined as 37 weeks gestation.
Ultrasound cannot accurately predict the size of your baby at term.
The other basis for early induction/delivery is that the placenta of diabetic mothers is thought to age faster. This is similar to the complications such as eye, kidney, etc disease but the placenta is much more sensitive. What happens then, is small areas of the placenta then start to die and detach. This increases your risk for placental abruption (placenta detaches from uterus) and other poor outcomes at 40 weeks. Think of the placenta like a funnel, capturing the nutrients and oxygen and delivering them to your baby. If there are holes in the funnel, fewer nutrients/oxygen go to the baby and if there is a large deprivation, this can lead to harm to the fetus.