Ruger LCP or LCP II in 380 auto caliber, 10.6 ounces unloaded. Very small. Very flat. It can easily fit into a ziploc bag or stretchy waist belt for runners.
You will need a permit to carry. Check the laws in your state. Go to a gun store, describe why you want a gun, and ask for recommendations. Most of the 9mm start around 18 ounces unloaded, even the small ones... plus, 9mm ammo is heavier than 380 Auto. That's too much to carry comfortably except in a backpack.
Carrying on the hip or a with a shoulder holster is impractical. You could carry the Ruger LCP or a light 9mm in the front chest pocket of some running vests. Look at the Sig Sauer P365 if you want 9mm and don't mind carrying an 18-ounce gun. It will probably weigh around 23 ounces when loaded.
Open carry on the hip or shoulder would frighten civilians and eventually you'd have to explain yourself to a cop. That could be very dicey. You could get shot. Concealed carry is the way to go.
Having said all that, you say, "I'm not much of a gun person." If you're not willing to take the training, get a permit, and learn safety protocols, don't carry a gun. It's a huge responsibility. Your running shoes are your best defense even if you're carrying.