Great question, hehe 😉
Short answer: Both, kind of.
A Little History
Short history of the Toolbar, ahem, Admin Bar: The Admin Bar was officially introduced with WordPress version 3.1.0 back in Februray of 2011. It generalized the existing Admin Header and made the whole thing more useful and more like a de-facto API under the hood. Since this new approach seemed to be a success it was overhauled after a few months already.
WordPress version 3.3.0, released in December of 2011, took the Admin Bar, renamed it to Toolbar and showed it also on the frontend – when logged-in only, of course. The Admin Bar grew beyond the Admin Dashboard interface and therefore its name needed to mirror this growth and extended functionality. This is pretty much the Toolbar that we still have today, where my Toolbar Extras plugin can bring a lot of extensions and tweaks to it.
Extension of the Admin
However, since its first introduction as Admin Bar, this little bar was and is considered just that, THE “Admin Bar”. Even after officially renaming it to “Toolbar” (for logic reasons), most users and developers still call it the Admin Bar. Which is easy to understand and to follow, as it will only show the logged-in users, it is generally considered an extension of the Admin Dashboard. And since the “Administrator” user role in WordPress has the most permissions and will show the most items within the Toolbar, it just feels as THE BAR of the Admin or for the Admin. So it all makes some sense.
My Plugin Work
For my plugins I have made the decision to use the terminology that WordPress itself uses. Therefore I use the term “Toolbar” since the release of WP 3.3+. As you notice, some of my early plugins still have the name “Admin Bar” in it – they were developed and released when the “Admin Bar” still was the official branding of this thing.
Call it Toolbar or call it Admin Bar, it doesn’t matter. What should matter to you is the question, how useful this bar is for you? Do you already use it as the central hub for your admin tasks – to get more productive and save precious time? If not yet, then check out Toolbar Extras now, it will power up your Admin Toolbar for you – to save you mouse travel and scrolling, to make your days in WordPress more productive!