Yeah, the new stealth errata is a bit clearer than the old definition of immunity, but still a bit vague.
You can have immunity to a type of damage (Acid or Lightning, for instance), to a type of effects (such as Charm or Sleep), or to more specific game effects (such as Stun or Forced Movement).
Only two immunities are treated a little differently: Poison (as that is both a type of damage and a type of effects) and Gaze (having immunity to gaze makes you immune to all damage and all effects caused by attacks with the Gaze keyword).
So, if you have Immune Fire and no Immune or Resist Poison and get hit by an attack that deals, say, 3d8+12 fire and poison damage, you take full damage as your immunity only protects from the fire damage and the damage is also poison damage, for which you have no protection. In regard to damage types it works just like Resist.
Similarly, immunity to effects (Immune Charm, for instance) only protects you from the non-damaging effects. You still take the damage, no matter what type it is, and if you're hit by a power with the Illusion and Psychic keywords that causes you to be dazed and take ongoing 10 psychic damage (save ends both), you're only protected from the daze as that's the only non-damaging effect. You'd need Immune Psychic to ignore the ongoing damage.
And lastly, immunity to a specific mechanic does not protect against anything else, so if you have Immune Stun, and you're hit by an effect that stuns you and deals ongoing 5 damage (save ends both), you're only protected from the stun effect and still take the ongoing 5 damage.
In other words, immunity to a type of damage simply acts as infinite Resist (unless it's Poison, since that also is a type of effects, and then it also protects you against effects with the Poison keyword).