iPhone provides my full name to other iPhone users - safety / security concern

It has been brought to my attention recently by two people that I have connected with on on-line dating sites that they both knew my last name because we both have iPhones and when I sent them a text message, iPhone gave them my full name. It wasn't the 'maybe?' functionality where my name/number was located in an email. Also, I checked that I DO NOT have "Share Name and Photo" enabled. I also experienced this when a guy from the site purchased a new iPhone a couple of weeks ago - I had him listed by his nickname in my contacts and when he sent me a message after getting the new iPhone, my phone alerted me with a message along the lines of "this contact's iPhone is listed under a different name" then gave me his first and last name and gave me the option to update the contact.

My phone default is to send new messages from my cell number

I do not have share name and photo enabled

These individuals have no identifying information about me except my first name and cell phone number

It is a huge safety concern for other iPhone users to have my first and last name provided to them if I am not ready to give that person my last name yet.