In the C++ language, which currently underpins the Arduino programming language, the Boolean data type is called 'bool'.
'bool' is a reserved keyword in C++, but not C, though the name 'bool', and 'true' 'false' are predefined in C in a header file called stdbool.h. The C type name for a Boolean is _Bool, which is a reserved keyword.
I just did a quick check, and both 'bool' and 'boolean' work inside the Arduino IDE, though bool is not highlighted, and boolean is. ('boolean' is defined in the Arduino IDE's wiring.h).
I'd prefer 'bool', as that is C++, and is available in C.
I sometimes write and compile code for the Arduino on my Mac, or Linux, so the more common it is, the easier for me.
I've never used the Bxxxxxx stuff. I use 0x... or 0bxxxxx.