Part of the confusion here is that key binding modes are not a one-to-one map with language modes.
When you created the language, if you clicked the "Inherit Keybindings and Callbacks" checkbox, then the base key mode for that language will always be used. Even if you pick that new language explicitly, it's still mapping back to the base keybindings, which in the case of Plain Text is "default". If you had created your language with C++ (or its variants) as the base language and checked the "Inherit Keybindings", you'd see that commands were being mapped to the "c" key mode.
If you do not click that checkbox, then you will be able to bind a key sequence to just that language mode (shown as "ext"). You'll still get key events and commands for everything in "default" (Ctrl+C will still copy) but now you can define your own bindings for just that language mode, which won't be pushed back into the default bindings.