How do I create a URL Button that, upon certain conditions being met, would save the record and send a notification

I'd recommend controlling the 'criteria' in the table notification itself.

i.e. When a record is saved and x, y, z are true.

It makes it easier to maintain. if this should happen everytime a new record is created.

BUT, if for some reason they may start it, need to save and return later, then I'd recommend creating a button that will check or uncheck a checkbox via the API_EditRecord function.   Then your notification will trigger off of that checkbox being changed.