Ayup, pretty good guess.
The wide bladed heads are for softwoods, where they can cut a wide and deep chip. The narrower blades are for harder woods. Weight of the ax is largely a personal matter, depending on upper body strength (not in delivering the chop, but for the wind up as once an an axe, like a sledgehammer...is started in the right direction, the weight of the blade/hammer does the work)
The curve of the blade, is also important. A very round blade tends to sink in very deeply in softwoods, as the leading edge cleaves the woody material. It also aids in rocking it out..however..it may sink so deeply that it never cuts a very good chip and you spend your time rocking it out. A blunter cutting edge shape cuts a shallower chip that is easily cleaned out by the next chop. In hard woods, a blunter narrow shape more commonly used.
The small short double bitted "cruising" ax was very popular with timber cruisers due to its short length, moderately light head weight and ease of carrying when on snow shoes.
The single bit axes have come in a bewildering number of shapes and sizes, from the heavy and wide Broad Axe, to the narrow barrel makers ax, to the modern firemans ax with its point on one end used for smashing open locks and doors, to the forest fire fighters Polaski, with an heavy duty blunt ax on one side, and a heavy duty hoe on the other, both sides equally used for chopping trees or roots in the dirt.
Actually quite fascinating, and then we can get into the various types of war axes....
Gunner
No 220-pound thug can threaten the well-being or dignity of a 110-pound woman who has two pounds of iron to even things out. Is that evil? Is that wrong? People who object to weapons aren't abolishing violence, they're begging for the rule of brute force, when the biggest, strongest animals among men were always automatically "right". Guns end that, and social democracy is a hollow farce without an armed populace to make it work. - L. Neil Smith