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She did send me a big message on mine after we broke up so now I'm thinking to either send a short message to keep the door open or just ignore her.
You're complicating something which means you care too much. Backstory aside, you're considering sending a text message for an outcome. If it were me, and the chick was cool at the end of the day, and it popped in my head that today or next week was her bday, and I felt like sending a happy bday, i'd just send it. I'd take 5 seconds send the text and be done with it. Same way I'd text a guy friend or a female friend. Because I FELT LIKE IT. Not to get her. Not to make her miss me. Not thinking about whether xyz would happen if I did or didnt. I wont think so hard on something simple. It pops in my head, I'd do it and be gone. I wont expect a reply or care either way.
Whether you do it or not is not the issue. The issue is it requires thought and a thread either way. You havent moved on then. Moving on does not just simply mean you're ignoring the person. It means whatever you do, you dont care.