To me, BASIC Engine (BE) is a platform to write and run BASIC programs.
The particular BASIC flavor of BE is Engine BASIC.
There are two boards, the original BE board, based on the ESP8266, and the Next Generation (NG) based on the Alwinner H3, this being more powerful, and easier to obtain.
So, you can run a BASIC program written in
Engine BASIC in the BASIC Engine board, or the BASIC Engine NG, though it will run way slower in the original BE (and some features may be missing).