Hello Sasquatch_10,
The indefinite (Turn Off) option when managing your notifications from the Lock Screen or Notification Center would apply to all notifications for Messages, and not a specific contact. If you chose this option, then you'd want to enable notifications for the Messages app:
"Change notification settings
Most notification settings can be customized for each app. You can turn app notifications on or off, have notifications play a sound, choose how and where you want app notifications to appear when your device is unlocked, and more.
- Go to Settings
> Notifications. - To schedule a notifications summary, tap Scheduled Summary, then turn on Scheduled Summary.
- Select the apps you want notifications for in your summary, set a time for delivery of your summary, then tap
at the top left. (See schedule a notifications summary.) - To choose when you want most notification previews to appear, tap Show Previews, select an option—Always, When Unlocked, or Never—then tap at the top left.
- Previews can include things like text (from Messages and Mail) and invitation details (from Calendar). You can override this setting for individual apps.
- Tap an app below Notification Style, then turn Allow Notifications on or off.
- If you turn on Allow Notifications, choose when you want the notifications delivered—immediately or in the scheduled notifications summary—and turn Time Sensitive Notifications on or off.
- For many apps, you can also set a notification banner style and turn sounds and badges on or off.
- Tap Notification Grouping, then choose how you want the notifications grouped:
- Automatic: The notifications from the app are grouped according to organizing criteria within the app, such as by topic or thread.
- By App: All the notifications from the app are grouped together.
- Off: Turn off grouping.
To turn off notifications selectively for apps, go to Settings > Notifications > Siri Suggestions, then turn off any app."
If you're not receiving notifications for a specific contact, we'll need just a bit more information. Is this isolated to Messages notifications? For example, do you receive call notifications for this same contact? Have you checked to ensure that the contact wasn't blocked: Block phone numbers, contacts, and emails on your iPhone, iPad, or iPod touch
We look forward to hearing back from you.
Cheers!
