Yes you can go out in public wearing just a diaper and shirt at 13 years of age.
A diaper covers you up and so there's nothing indecent or wrong about you going in public wearing just a diaper.
That's perfectly fine for you to go out in public wearing just a diaper even at 13 years of age.
If you're more comfortable wearing just your diaper in public then that is all you should wear when it's warm out.
I've seen many other older kids in diapers in public as well as 8 year old boys wearing just diapers.
If toddlers and babies can go out in public wearing just diapers then so can older kids, teens and even adults.
When I was a toddler I was dressed in just a diaper and shirt and also during my Kindergarten years my mom did put me in diapers for the rest of the summer because I had to wear diapers during the school year because I wet my pants.
So during the summer when I was 5 years old and almost 6 years old my mom dressed me in just a diaper, shirt and shoes.
I went in public like that to the store, doctors, etc and at home I was outside playing in just the diaper and shirt and shoes.
It felt great to me and I actually loved it.
When I have kids I"m gonna dress them in just a diaper and shirt and shoes and take them in public.
At home when it's warm they are gonna be in just a diaper and shirt and maybe socks just like the diaper boy Tommy below.