I see no difference in 1 or 2, except that 1 is a bit more clear in implying you'll call one or the other, but not both.
To call, meaning to telephone, versus to call, meaning to be known as, is one of the keys to solving your other question.
What is the difference between 1 and 2? Are they interchangeable without any difference? Sometimes "or" has the meaning of "that is to say, in another word" like in 3, so I doubt if ex2 can be interpreted as ex3 like "him=her".
ex1) I will call either him or her
ex2) I will call him or her
ex3) This is called HD, or High-Definition TV.
I see no difference in 1 or 2, except that 1 is a bit more clear in implying you'll call one or the other, but not both.
To call, meaning to telephone, versus to call, meaning to be known as, is one of the keys to solving your other question.
I'm not a teacher, but I write for a living. Please don't ask me about 2nd conditionals, but I'm a safe bet for what reads well in (American) English.
You helped me understand "either" can play a role to distinguish more between two things, and my last question was not related to the meanings of "call", but another meaning of "or"= "that is, in another word"
ex3) HD, or High-Definition TV is getting popular.
ex4)He or Mad Dog was notorious for hitting students' faces with his slipper.
(In this, he was a high school teacher who used to hit students, so was nicknamed Mad Dog)
Number 4 doesn't work- it could work with a name, but with a pronoun, it sounds as if Mad Dog is not the same person. Also note that the punctuation has changed in number 4.