I'm not sure which one to use: I'm "coming" or "going" as I promised.
And why would I use that specific word instead of the other?
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It depends on your relationship to the place where you are and where the recipient of your message is. If, for instance, you are at work and there is a meeting you are both expected to attend, then you would say, "I am going to the meeting as I promised." If your colleague is already at the meeting and waiting for you, then you would say, "I am coming to the meeting as I promised." Where are you in relation to the place? Where is your recipient? You would tell a colleague at work, "I am going home now," but if you are on the phone with your spouse who is already at your home, you would say, "I am coming home now."
So in each case, ask yourself, "Am I coming or going?" in relationship to the recipient.