An arpg is an rpg with action combat. Its a huge wide open lable. Any rpg which uses turn based combat and a hex or square field is called a tactical rpg, even though games in either genre can differ wildly. Diablo & co. are often called hack & slay games, since they are dungeon crawlers. The first computerized rpgs (like wizardry and the very early Ultimas) were also Dungeon Crawlers, so those are rpgs, in that many pen & paper rpg adventures like Dungeon and Dragon's infamous Tomb of Horrors was nothing more than a dungeon crawler. I find it amusing that people (including Josh Sawyer) don't consider these dungeon crawlers rpgs, since they have no branching dialogue or story decisions, but there you have it, they were the first rpgs, both in terms on pen & paper, and on the computer. Of course there are also open world rpgs (both tactical or action) and linear story based (both of these also have a tradition in pen & paper games) but the word ARPG refers solely to combat, not how the adventure or the computer game is structured.