Modes are the core concept on which Vim's design is structured. They organize both the interface and the user's mental model of the text editing process. I did a quick case-insensitive search over Vim's help files, and they matched 'mode' 4470 times.
The first thing you must learn (to use Vim) is that it is modal. And this detail really tends to stand out when first meeting Vim. I think "modal" is a nice word to concisely describe and differentiate Vim.
There are 13875 matches for mode in the Emacs docs.
Modes are even more foundational to Emacs than to Vim. A new user will encounter a dozen modes by simply starting to use Emacs for a few minutes. Every buffer has a major mode, and there can be many minor modes enabled per buffer or globally. The entire editor is built on a tree of modes, all inheriting from the aptly named Fundamental Mode.
At a basic level, each mode affects what every key does. The idea of keys doing different things in different modes is not unique to Vim.
Which just goes to show that Vim's definition of "modal" is somewhat contrived, as Vim's definition of "modal" applies to a very specific implementation of two to four modes (which coincidentally Emacs also offers multiple versions of, both natively and as third party packages to varying degrees of faithfulness).
Emacs uses the word "mode" to mean something different than Vim does.
An Emacs "mode" corresponds more closely to what Vim calls "plugins" (Emacs's "major modes" are like Vim's "filetype plugins", and Emac's "minor modes" are like Vim's other "plugins").
Vim's modes are more foundational in the sense that all Vim plugins are structured and organized in terms of the same set of modes. The core of Vim is the modes (insert, command, etc.), which are then customized. Emacs calls the customizations themselves "modes".
I don't consider Vim's definition of "modal" contrived, and I don't consider Emacs to be "modal" in the same way (though you can of course customize it to be).
The first thing you must learn (to use Vim) is that it is modal. And this detail really tends to stand out when first meeting Vim. I think "modal" is a nice word to concisely describe and differentiate Vim.