For Test cricket:
2 more meaningful categories:
Batting all-rounder - picked for his batting, and can also bowl test standard
Bowling all-rounder - picked for his bowling, and offers significantly more with the bat than a "true tailender"
Bowling all-rounder is a bit more "valuable" of the two, and seen more often in Test cricket. The value of the batting all-rounder will vary based on the quality of the front line bowlers and the front line group's tolerance for a heavy workload. A good example of this lesser value for batting all-rounder I think could be seen with Shane Watson back in the Australian side's heyday under Punter. His ability to send down Test-class, but middling quality, seam up overs was mostly an afterthought, and he needed to have excellent batting form in Australian FC to get his chance in the Test side.
Less important category:
True/"genuine" all-rounder - More of a trivia category for mine. I'd define it as a player who while still fitting into one of the categories above, is better or arguably just as good as specialists in the side. So for the quintessential "balanced" example of Ian Botham. He was picked in the side for mine as a bowling all-rounder, but there were significant stretches where he was one of the top 3 batsmen on the side, and for that reason he merits the title "Genuine" all-rounder, but of course this one is a bit more subjective, as to who earns the title. Shakib al Hasan is another great example.
Secret category 4:
"Bits and pieces" cricketers - clearly don't merit a place in side for either batting or bowling, but somehow do enough combined that the selectors keep giving them an extended run in the side. IMO, selectors need to **** off with these kind of picks for higher level Test sides. Ashley Giles is the classic example.