What is the harm of finding a new job and leaving? Is it better to suffer for 1 year working for a toxic person or to harm my reputation by leaving so quickly?
It depends on your sanity and state of mind. If there are positives in the role you can focus on and enjoy, stick the role out. There is probably a lot you can learn, even if it is just how a non-profit operates.
However, if this starts to get out of hand (e.g. you lose sleep, it affects relationships with others), get out as soon as you practically can. Your health and relationships are not worth it.
What is the best way to explain why I am leaving, during interviews, without making myself look weak or like a "runner"?
Everyone is entitled to one or two "wrong fit" positions. When asked, just say "It become quite clear they were looking for someone else". If pressed, say something like "They were clearly looking for a more junior staff member and I wanted something more challenging".
The critical thing is do not bad mouth your current employer. No matter how justified or correct you are, it is very difficult to not make it sound like you were the one being problematic. The example above both explains why you left without making them sound bad.
If you do "vote" for me to stick it out, what are tangible/applicable ways to handle such a boss?
Bosses come in all shapes and sizes. Sometimes bosses create opportunities for you to work through them and improve together. Other bosses create opportunities for you to move into the skill and strategy gaps they leave.
Even if you do not believe it will work, I would first talk to your boss about these issues in private. Start with some facts (e.g. "Yesterday person A said this. You said this in response. Today person B said this and you said a similar thing to yesterday.") then give your inferences (e.g. "It sounded like you were talking down to them) and ask her side.
Keep in mind that you may be wrong or there may be good reasons for the boss's behavior. Regarding the employee interaction, maybe that employee has had performance or truancy issues in the past. Regarding the apparently lack of vision, maybe the problem is higher up the management chain (personally, I have found this almost always to be the case when vision is unclear or constantly changing). If you go into the discussion certain you are right and ready to yell, the meeting will be a disaster.
If you can do so, talk to others in the organization. You have to pick the staff member (preferably one with lots of experience and one that is not too emotional) and the time (not just after an argument). Contacts in support departments like HR and accounting can be great for this.
Try to be positive and focus on improvement rather than point scoring. In the employee interaction example above, suggest a different phrase the manager could have used talking to the staff. The idea is to suggest this without implying the manager sucks. No manager wants to be disliked and every manager wants to manage more effectively.
Similarly, be aware that your own performance and attitude may need improving. You may have been a superstar at your previous role with bucket loads of experience but each role is different and requires different things. You may need to unlearn things or do things the "wrong way" to see how they go. You may still believe it is the wrong way but you may learn a thing or two.
Lastly, it is very easy to focus on the negatives of a certain situation and much harder to focus on the positives. Write a list of all the good and bad things about the role, as objectively as possible, and share this with a friend in the same industry. They may help you with the relative importance of these points. In six months, no matter where you end up, read back through the list and see if you agree with the relative importance of each. Even if this does not help you now, it can point out things to avoid in the next job.