Is “or” exclusive?
From your first example
A or B mean one but not both?”
This is a question which has raised some debates both here on ELL and English Language & Usage Stack Exchange.
The short answer is “it usually means just one, but sometimes it can mean both are allowed.” It depends on the context.
The default reading you are using is correct, just one of them needs to attend. (But perhaps both of them will— it depends.) See here for more on this topic at EL&U.
Note: because “or” can be ambiguous in this way, in the study of logic, and in several programming languages, the term “XOR” is used for “exclusive or” — one but not both.
Your sentence is a very brave attempt at solving a hard problem:
🙁 At least either she or he should attend
Unfortunately, at least either does not work very well and sounds awkward.
The sentence could be rephrased this way:
✔️Yes: At least one of them should attend.
✔️Yes: At least one of them, but not both, should attend.
✔️Yes: Between Alice and Bob, at least one of them should attend.
✔️Yes: Between the two of them, at least one
(of them) should attend.
✔️🤩 Yes: Alice or Bob, or both of them, should attend.
The last one is the simplest and best. Notice that it uses “them”, but it also establishes who “them” is (who “they” are).
If you can’t use “them”, I would use their names instead of “she and he” because this still sounds awkward:
😐Acceptable but awkward: Between she and he, at least one should attend.