Both offer some code and insight for the HD44780 and serial in general that I thought was good. Not that I think you just could copy/paste the code and voilà, it works, but maybe they can start you in the right direction? Hope it helps, sorry I don’t know too much about your SLCD.
The reason that display was so cheap is because it does not have a “serial backpack” so you can’t send it commands like you did with your picaxe lcd. This is covered in Manual 3, Page 38.
As it is 44780 compatible you should be able to use it in 4 bit mode, which will require 6 pins of the micro. If you are really brave you can learn to hook it up w/ just 2 wires, using just a diode, resistor, and a 74LS174. I found someone that had written a library in basic that I then converted to jal. I don’t know if my conversion will work yet.
After reading your post about the AXE033, I am under the impression that the suggestion I made could be outside of your skill set currently. I am not saying that I know what I am doing either. I am just saying that you will likely need a good deal more assistance than I can offer.