Action not updated

On the States section of the Developers tools I can see notify.mario_mobile

If I try to test it in the Actions section, I can't see that state, but I still can see the previous one, which was the name of the device. Why it doesn't update?

You are confusing Actions and Entities. The action would be notify.send_message with notify.mario_mobile as the target:

action: notify.send_message
data:
  message: Hello World
  title: My First Time Using this Action 
target:
  entity_id: notify.mario_mobile

Currently, the entity-based option notify.send_message only supports basic notifications with just message and title... notifications that use more complex data will still need to be handled using the notify.mobile_app_* actions.

Thanks so much. That actually worked.

Yes I'm very confused because both different things are called 'notify'. So is the 'notify' in the action the name service? What does it mean in the entity_id and where is it coming from?

Also, why in my second screenshot there is not mario_mobile but it's previous name instead, why it doesn't update? (I updated the ids in the mobile device)

In all cases, the notify refers to the underlying Notify integration which is a "building block" integration.

If you want to change the service ID, I think you have to remove the companion app from your phone and delete it from the integrations settings, then change the device name and reinstall the companion app... but I wouldn't bother doing that. With the way the service ID is created, you will never get an action notify.mario_mobile... I'm pretty sure you will end up with notify.mobile_app_mario_mobile.

If you want an action notify.mario_mobile, use a notify group.