I was looking for the answer to this question because I have the same problem. I'll try to explain it better. When you are in your messaging app (or in handcent), when you see a text message conversation the message from someone else has their name. A message from you to them has "Me" as the name. I realize it doesn't show that way to the person I sent the text to, but on my screen, text messages from me have "Me" instead of my name, for instance.
So let me get this straight..
On my phone:
Me: nadda.. where's the bathroom in this thing?
On DrWho's phone:
ThirdWave: nadda.. where's the bathroom in this thing?
Sending a text it should only show as "Me" on your phone.. on the other phone it should show what they have you as in their contact list.
Having your own name show up on your phone is a bit redundant imho.. unless you really must refer to yourself in third person.. in that case not seeing your name by your response on your phone is the least of your worries.. lol
I am having the exact same problem. My sister gets an e-mail from me and it says "ME", not my name. But my hsuband gets my name. My sistre also gets two of every e-mail, but my husband only gets one. If I send an e-mail to myself, it comes thru from "ME" and I get two copies. This is a real problem if I am sending an e-mail (for business) to a new person