how do you deal with it when you cry easily in inappropriate situations even when you're not that upset about something?

While not the same situation at all, I personally, for myself, put the expectation that I never cry around anyone. I don't know where this came from since I was raised in a very open family where emotions were encouraged. That said, I cry very easily when the anxiety of people seeing it is gone.

I met with my psychiatrist today, we talked about my mood changes. When I told her I spent most of my day in bed, trying to avoid life, I suddenly broke down. Like voice-cracking, insta-tears, trying to do math in my head to stop it, crying.

I was sure that she thought I was just feeling sorry for myself and that it's not real depression. That I'm a big phony.

I knew, logically, even when I was standing up there, that they were asking me things I had no ability or obligation to know.

Rather than looking at this as a flaw in you, look at it in a flaw in the people asking the questions. They set you up to fail those questions because they weren't prepared. This isn't on you.

Great, now they all know me as the girl who can't take critique.

Some might, but there will certainly be people who understand it's only human. Some people who will empathize that it's not about you not being able to take critique, but that your crying was a SELF-critique on your ability to answer the questions.

You're also 23, yes, an adult, but you're still young. Deep down everyone knows that early twenty-somethings are still figuring out modern life.

Btw,

A professor pointed out I'd put a picture of the wrong bird on my slide, which was awkward, but whatever.

Don't forget that this is your real ability to take critique. You know you made a mistake, you accepted it and moved on.

If you have the chance to, I would get a meeting with the professor to discuss the presentation. Double-check what your educational responsibilities were. It wouldn't hurt to drop something like, "I felt overwhelmed by the questions I wasn't prepared for. I am concerned that I wasn't adequately prepared."

At worst, you're going to be, "the student who cried", and you wouldn't be the first, last, or even some exception. Nobody is going to judge you as harshly as you are right now.

Learn to stop mentally bullying yourself when you feel out of your abilities. Not knowing something isn't a flaw when you're actively learning.