There isn’t a ‘Today’ perspective by default in OmniFocus. Are you referring to the Mac and iOS today screen widgets? In that case, what appears there is customisable in the OmniFocus settings/preferences with the badge customisations.
Huh, that’s odd; I have one in both OF Mac and OF iOS – they even have their own icons (a calendar page) and in the case of the Mac, keyboard shortcut! I have uploaded a pair of screen shots. Where do you think they might have come from?
Not sure where it came from, but it’s a custom perspective. If you click on the eye you can adjust its settings, or from the Perspectives editor window.
As for why you see different items on Mac and iOS, it seems that the Mac version of the perspective has been changed from the original settings (the text in the side bar is italicised and the eye icon is different. If you click on the sidebar icon again or click ‘Revert’ from the eye menu/view options, you should start seeing the same items on both Mac and iOS.