Here is the classification that I use for Hammers and Ice Axes:

• If it has a hammer face and has no spike at the handle's base, I call it a "Hammer."
• Otherwise, if it is modular^†, I call it a "Technical Ice Tool."
• Otherwise, if it has an adze, then I call it a "Traditional Ice Axe"..
• Otherwise, I call it a "North Wall Hammer."

^† "Modular" means that it has an interchageable adze, hammer, pick, and/or spike.