What controls the text preview in notifications?
Everything in the notification is set by the app that created it. So in the case of your email, it's the email client.

Note that in recent Android versions, you can expand most notifications by pulling down on them with two fingers. In Gmail, for example, if you have a notification about several emails, you can expand it from "3 new messages" to show the sender and subject line for each. If you only have one new mail, the default layout shows the sender, subject line, and first line, but the expanded form shows a few extra lines of the content.