For camping I would recommend a full size single bit axe (about 36" handle length, minimum 3 pound head). That will handle whatever needs to be done in terms of real chopping and limited splitting.
You will most likely be using smaller, dead wood in your fires. For that task, a bowsaw (or better yet, a small chainsaw) will cut small logs and sticks easier that an inexperienced person can ever do with an axe. Dead wood is much harder to cut than green wood.
Hand axes or "hatchets" are just that....meant for lighter duty work. They lack the weight and the handle length to develop the kind of power stroke needed to perform real work.
Pulaskis are a sorry excuse for an axe. These were not developed for chopping wood, but for digging in the dirt and grubbing roots in order to construct fireline. They are clumsy to use for chopping as the balance is terrible. In the hands of amateur, they are more hazardous to use for chopping than is an ordinary axe, so they should be confined to use as a digging, rather than chopping. Accidents involving pulaski use by experienced users are very common and, in my opinion, are inherent in the tool's design. Like most "combination" tools, they do both their jobs so-so and are inferior to a tool designed for a sole, specific purpose.
Double bit axes are for "professionals", or those who use the tool to make their livings. In the old days, that included farmers and homesteaders. Given that most professionals use power saws now, they are becoming obsolete, except in congressionally designated wilderness areas, where power tools are illegal to use and down logs in trails must be removed by crosscut saws combined with axes. One edge is honed fine and to a steeper angle and is used to take out the clean meat of the log and to bite exceptionally deep. You can easily shave ribbons from a piece of paper with the fine edge of this tool. The second edge is sharpened to a blunter angle to allow the cutting of bark and knots. This edge profile is resistant to damage but will work you to death. Bark, and bark with dirt, will dull a fine edge fast. Knots can damage a steep angled finely honed edge in a hurry. Thus the two edges.
