Was I too eager and desperate in my interview follow-up/clarification email?

I had a second interview today (first with the actual manager) for a position with an American company. The role would be remote from any American city. I’m Canadian and mentioned to HR in my first interview that I could initially work remotely from Canada for a month or so while my lawyer & I get my TN work classification application in order and then while I look for an apartment. Well I stupidly forgot to mention this clearly to the manager in this interview. I did tell her I’m in Canada and will move once I have a job offer that’s a good fit (I’m currently working as a contractor for an American company remotely which I started last month but I wouldn’t move for it as the role was misrepresented to me which is why I’m applying to jobs and I told this to the manager. Side note- I wouldn’t have to give notice which plays in later). So at the end of the interview, which went very well I think, the manager told me that I was really great and had great experience and to email her if I had any questions at all. She then said she hopes I have a happy holidays (which I found a bit odd since it’s only Nov 30). After I told my mom how it went well, she said I should have clearly told the manager that I can start immediately from Canada in the meantime. It’s possible that they’re not looking to start someone in December so the happy holidays thing wasn’t very odd, but that I should still email her to clarify that I don’t need to wait until I move to the US to begin working. SO I sent her a short email thanking her for her time and clarifying this point. But I guess since I was anxious to really clarify, I italicized the word “immediately” before “if needed”. At the time it made sense in my mind but just now I looked at the email and feel stupid. I don’t expect her to reply to this email because I’m sure she would still need to talk to HR about me as a candidate in general, but have I hurt my chances by possibly sounding desperate?

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