If my first email contains everything required for you to locate and fix your security issue, my motivations are pretty clear.
"I'm an independent researcher who posts under the name JayeLTee. I look for publicly exposed data online on my free time and alert the companies affected to try and close the exposure."
There is nothing more than that, want me to make a fairy tale story to tell the companies? I try to be as clear as possible and pass the message as clean as possible with no BS on the email, again because I'm not selling a product or a service.