How do I change my Mail program to IMAP so it deletes mail on all devices?

Actually, it wasn't very helpful. I was hoping "helpful" was a toggle with a "less than really helpful" setting. my bad.

Everything about your post is confusing, including the high tech gif which doesn't tell me where to begin. thanks for trying. I'm back to square one.

For instance, I do not see "Passwords and Accounts" under settings. I only see Passwords.

It turns out the solution was much simpler. In the past it had to do with POP vs. IMAP, according to some threads I read. Now the problem is that Mail seems to default to putting "discarded" messages in archived messages.

For any future person having this problem, go to mail, then accounts, then select the account (this part is not obvious, but you tap on the actual account), that brings up the account information, then go to advanced (also, not obvious) - then under "move discarded messages into," select deleted mailbox.

Why apple programmers thought it would be helpful to have discarded messages be something different than deleted messages is beyond my understanding. It's been very annoying, and the fix is far from easy.