The rules of ARAM are always up to the host, but you are giving a very inaccurate depiction by claiming that Blue Pilling is not allowed, Snowball items are not allowed, and Fortify is not allowed. If anything, the few games that do not allow such things are usually referred to as Nerf ARAM or Carebear ARAM.
Yes, most of the rules are up to the host, but most of them are the usual: "No snowball, no fortify, all random, all mid, no recalling, no buying until dead." I have never played a A.R.A.M game where you can recall to heal since the point of staying all middle lane is to grab kills and assists. There would be no point in laning middle lane if you can have the ability to recall every 2 minutes because you were "low on health." You might as well play a normal game for recalling.
Also, some people have mixed feelings about most of the rules I have put up. I agree with them since every game is unique to the host. I'm just putting up a general rule for A.R.A.M players since most people don't really know what it is. Of course, when I first played it, I had no idea. I like the idea of "no recalling/blue pilling" and "no snowball items" in the general A.R.A.M because some champions that are picked randomly have a more Ability Power orientation and can have the upper hand in the game.