I think one thing to note about Fairy Godmothers is that it's family run, not corporately run. It has never been officially sanctioned by Disney. (I think it was Kindercare before Kids Nite Out.) And there is a lot more formality to corporately run places. Also, the woman who runs Fairy Godmothers has done so for 25 years plus, and so she's probably used to doing things more casually than has become the norm these days.
Both our Fairy Godmothers showed up with a suitcase full of toys, but no real "paperwork" to speak of other than Fairy Godmothers fliers and cards. Both were early, too.
Kid's Nite out sounds good, too. But I'm not sure that the poster's situation would have been much different with them (the rain could have made the sitter late, prevented them from bringing up the toys, she wouldn't have heard the phone in the drawer, etc.) But that would have had paperwork (I think there's some stuff to sign) and they would have been in uniform with a nametag.
If you're nervous about any baby sitting situation, I'd consider having dinner at the resort you're at, that way you can check on the situation firsthand several times if you like. (We did that the first night with Fairy Godmothers. ) Or at least flag them that you'll be calling and to expect it, and to answer the phone!