Certainly a 2" diameter is going too large for most hands to hold. I've looked over my sticks, and none have more than 1.5", most closer to 1"
I usually press on stick stock with a little sideways pressure as well as downwards. If the stick flexes more than a little, I don't bother with them. Ones that flex slightly are OK by me for walking around town. But if I want something for trekking up and down hills, or supporting me when hopping over streams, I want something very sturdy.
Part of the question can be answered by the kind of wood. I've managed to find 1 good piece of hornbeam. The grip of the stick has a cross section of about 1" x .5", which is almost too thin for my stiff hands. The wood, however, is completely rigid. I have a few of white oak and one of hickory, and the cross section on those is slightly less than 1". But I also have some made from sassafras and sycamore which are a bit thicker, as those woods are much lighter and less shock resistant.