A co-worker who got hostile over being told your real name sounds like a nightmare.
As a former secretary, you should absolutely correct yours. Its kind to do it in a neutral way, but she should be mortified.
If you've had to correct her more than once, the "check your autocorrect" is nice for the second instance - unless it was communication to a third party.
If she's spelling your name wrong anywhere another person would see it (or g-d forbid, to clients), that is unacceptable and she should be told firmly that calling *your boss* by the right name is a minimum job requirement.
I can imagine situations where a long time employee might be supporting a group, and if you were new to the team she might get mixed up. But otherwise, that's pretty ridiculous.
By the same token, a vendor can be told quite matter-of-factly "it's Melissa, actually. Thanks."
If they do it again, I'd tell them, "That's the second time you've gotten my name wrong. Please make sure you update your records." If they can't get it right after that, it's valid for that to color your view of their competence to deliver whatever you hired them for.
For clients of yours, you really can't escalate unless they're problematic in other ways. If they can't/won't get it after 3 low-key corrections, I might just suck it up as long as they pay their bills on time.