it will show pushname and be delivered to all in group-convo; you want a private chat you hold on to the name to do so...

Its a feature so in group convo when messages from multiple parties are being sent/rec'd its easier to differentiate who you are intending the message to...

Pushname is better than contact name as in group convo everyone has their pushname so imagine different people saving the person differently in their contacts; its going to be a mess when replying...