I am not positive, but I believe “Ax” is a generic way of mentioning TinyFPGA A1/A2 (XO2-640/XO2-1200 FPGA) and “AX2” may be just be a typo in that table heading (board pictured says “A2”).
I do believe the older B1/B2 boards have been superseded by the newer Bx boards.
At any rate your A2 board should be totally fine to use (and good for learning). You just need to download the Lattice Diamond software (and get a free license). Hopefully you already have a TinyFPGA programmer (and the upload utility for it) or other Lattice Diamond compatible programmer (e.g., FTDI 2232H breakout from Adafruit or Seeed).
Here is a small project I made showing a version of “Ben Eater’s breadboard computer” (https://eater.net/) modeled in VHDL (a bit more than a blinking LED, but not too complex). That project has a TinyFPGA A2 project file that should be ready to go. See https://github.com/XarkLabs/BenEaterVHDL
Lots of other VHDL and Verilog examples to play with out there if you search.
Have fun, and feel free to ask questions here if you get stuck.
P.S. In my experience, other than having to configure the input/output pins, clock and I/O devices for each board, most VHDL and Verilog designs on the web are quite compatible and usually not too hard to “port” (although some can use “vendor primitives” that you have to replace).