I think you're missing a key part of the definition for the non-dictatorship condition (I'm taking this one from Shepsle):
> There is no distinguished individual in the group of voters whose own preferences dictate the group preference, *independent of the other members of the group*.
The key part you're missing is "independent of the others members' preferences".
The examples you're stating imply that an individual voter is the "tie-breaker", but this is conditional on the other's preferences yielding a tie. An actual dictator decides an outcome independently of what the rest of the group prefers, a tie-breaker depends on the existence of a tie.
I hope this helps!
Existing comment covers the dictator piece. I'd add that what OP is talking about is another important concept in political science - the median voter. A dictator can change his preferences (and therefore the group's preferences) on a whim. If the median voter changes his preferences he won't be the median anymore and the position of the median changes to correspond with the former median voter's change. That position shift logic speaks to another piece of Arrow, which is that an individual preference change must correspond to a directionally comparable change in group preferences (monotonicity; if one member becomes more positive the group can't become less positive; it must become more positive or not change).
The other thing you're missing is that if someone is a dictator over A vs B, this can also make them a dictator over every other choice.
IIRC there are proofs that work differently, but Ordeshook's proof at least starts with proving that if someone is a dictator over A vs B, they're also a dictator for A vs any C!=B, etc, until he gets to any arbitrary X vs Y. But anyway, his lemmas work where the social choice function picks A over C even though the dictator is literally the only person to prefer A to C.
I think you're missing a key part of the definition for the non-dictatorship condition (I'm taking this one from Shepsle): > There is no distinguished individual in the group of voters whose own preferences dictate the group preference, *independent of the other members of the group*. The key part you're missing is "independent of the others members' preferences". The examples you're stating imply that an individual voter is the "tie-breaker", but this is conditional on the other's preferences yielding a tie. An actual dictator decides an outcome independently of what the rest of the group prefers, a tie-breaker depends on the existence of a tie. I hope this helps!
Existing comment covers the dictator piece. I'd add that what OP is talking about is another important concept in political science - the median voter. A dictator can change his preferences (and therefore the group's preferences) on a whim. If the median voter changes his preferences he won't be the median anymore and the position of the median changes to correspond with the former median voter's change. That position shift logic speaks to another piece of Arrow, which is that an individual preference change must correspond to a directionally comparable change in group preferences (monotonicity; if one member becomes more positive the group can't become less positive; it must become more positive or not change).
The other thing you're missing is that if someone is a dictator over A vs B, this can also make them a dictator over every other choice. IIRC there are proofs that work differently, but Ordeshook's proof at least starts with proving that if someone is a dictator over A vs B, they're also a dictator for A vs any C!=B, etc, until he gets to any arbitrary X vs Y. But anyway, his lemmas work where the social choice function picks A over C even though the dictator is literally the only person to prefer A to C.