The attached picture shows an ax about the right size for most city folk. A full size ax is a bit heavy in the hands of a beginner, or a smaller person. A hatchet is too small to fell trees or split logs. Because we are discussing a tool to be carried in a truck, there is no reason to have less than something optimum.
I carried a woodsman's friend on my sailboat on a circumnavigation. It was almost useless, more of weapon than anything else. I think a light tool like that is best in a tropical setting as a substitute for a machete, useful for clearing brush around a campsite. For camping in our temperate climate, minimal tools include an ax, a paring knife, and a pair of Vice-Grips. A camper who knows how to keep an ax and knife sharp probably knows how to use them, and vice versa. Regardless of the tool, the skill of the user is more important than the tool itself.