Perhaps different wording will help.
Rare is a hiring and in play restriction. As long as the dreamer is in play, Teddy(s) ignores this and has the Nightmare characteristic.
If the Dreamer dies, then Teddys in play are still nightmares and still ignore the Rare 1, but new Teddys wont have it, so they cant be summoned by Nightmare summoning (not sure who has that...) or if there is already a Teddy in play (due to having the same name and Rare1). If there are no Teddys on the field and one gets brought into play after the Dreamer is dead, then it won't have Nightmare and will be Rare1.
Also, remember that buried models are still 'in play' and don't get summoned, they get 'unburied'