In response to "Have it so the name in the list change so people know which role your Egi is" I think "The name isn't in the list for other people." is a perfectly valid argument. That said, unless the person is using the Tanking Egi, yes, I suppose it doesn't matter which Egi they use in most situations.
Assuming the three Extra MSQ Primals from 2.0 have Egis that map directly to the three roles, I'm not sure which between Shiva and Leviathan fit better for Tank or Melee. Ramuh is however, basically a shoe in for Caster.