Neither and either

[Pipbin]

Am I wrong in thinking that they are two different words?
In my understanding either is a choice, 'you can have either cake or ice cream'.
Neither is when you don't want any of the options or you are agreeing with someone else's negative.

'Would you like either cake or ice cream?'
'Cake please, I don't like ice cream.'
'Me neither.'

Yes, they're two different words; your examples are spot on. And a true pedant uses "or" and "nor" in the same way - 'I have no cake nor ice cream in the house - would you like a hobnob?'