You can summon anything if you have enough aether and enough faith/willpower/something.
For Dalamud, the goal of Louisoix wasn't to summon "the twelve gods", but only to gather a lot of aether and for that he needed every eorzean to focus their thought on the same thing, and it just happens that the concept of "the twelve gods" is something shared by all Eorzea.
"Primal" is just a name for anything summoned, but it also has some political meaning : we believe in gods, it's a good word so we're good, and they believe in primals so they're bad. ... even though Ifrit, Ramuh, Titan, etc. are simply the gods of each beastmen tribe.
And, that being said, I wouldn't call Shiva the goddess of a people. She looks more like the boss of a band of thugs. They just share a very strong belief in their ideal.
And there is this someone who crossed all Eorzea and helped all of its inhabitant with big and small tasks. I wonder what could happen if they all prayed for him/her...