The half-moon icon on your iPhone means you've enabled Do Not Disturb, or muted a specific conversation in Messages. Here's what you need to know.
When "Share Focus Status" is enabled on your iPhone or iPad, anyone texting you will be automatically told you're using a Focus mode.
This article explains how to turn off the Do Not Disturb setting of the Focus feature on an iPad. We also cover methods for turning on this setting. Swipe down from the top right of your iPad display.