Winged Dragon of Ra - Phoenix Mode VS. Monster with "Unaffected by Card Effects"

This may be a question asked to death at this point but I wanted to get a specific ruling information about this. Yesterday, I had the normal Winged Dragon of Ra on the field, my opponent summoned a monster that was "unaffected by card effects". I activated "Sun God Unification", sending Ra to the graveyard and summoning "Winged Dragon of Ra - Immortal Phoenix" from the graveyard. I don't remember what my opponent's card was but it was "unaffected by card effects" + "detach one monster and send one card your opponent controls to the graveyard". It successfully killed my immortal phoenix, which I thought was impossible.

Then the next turn, I summoned another Ra and with the help of "True Sun God" I was able to summon phoenix mode again. I used its special ability, paid a 1000 life points, but I couldn't send that monster to the graveyard. How is it that both monsters are "unaffected by card" effects, but that monster is more unaffected somehow? Is it related to summoning order, like with Skill Drain? Whichever "unaffected by card effects" monster gets summoned first gets to remain unaffected? Anyway, I'd really appreciate some insight into this. Thanks!