Are they dynamic because the content of the label changes or dynamic because they turn on and off at different scales?
A couple of ideas:
Is it a map reference scale you are looking for? That's where the software fixes the size of symbols and text to the desired height and width at that scale.
"annotations via a service" - You can only share annotation features as a service if you have ArcGIS Enterprise 10.6 or higher. That is new functionality. Are you currently using that version?
You can also create label classes that turn on and off at different scales, then publish that feature service.
If you create a feature-linked annotation then when an attribute of your original data is changed, the linked annotation gets updated too. Is that what you are looking for? When the tool you mentioned above opens, make sure you check the box at the bottom for Create feature-linked annotation.
You might start looking at using Arcade to create some dynamic expressions across the platform.
If any of these ideas are close, let me know and I can elaborate and help you out.