Remove uninstalled applications from the Notifications system preferences
I can't say for certain if this trick will work on older macOS versions, as I no longer have access to them. But, in High Sierra 10.13, you can just left-click once to highlight the offending entry, then tap the <DEL> key (not backspace, forward-delete) to remove it.
I have two entries too, but they are not old and new. One is Twitter App, the other is system "build-in", this Twitter account for share(for example, click share button on Safari you can see it) and many other things.
You can check, "System Preferences"->"internet account".