Are you continuing the same message conversation with this person that you started on your iPad (which would have been identified with your Apple ID)? If so, you will need to start a new conversation with this person from your phone for it to be identified with your phone number (and since the recipient has your phone number in his/her contacts it will be identified by your name).
I have the same problem. When sending messages from my MB Air the majority of the time they get truncated losing the last one or last few words. On the MB Air the messages appear with all the text, but those who have received the messages say they are being truncated. I can confirm this to be the case as when the messages are sync'd to my iOS devices they are also truncated. Sending messages from iOS works fine, so it seems to be an OS X issue.