Hi,
I think you need more than a "Code Snippet" because the Orbital LCD looks to be an "intelligent" LCD (at $40 ?), whilst the "few $ 1602a for Arduino" is probably a very "dumb" LCD, dependent on a complex library for the Arduino. It
is possible to drive such displays with a PICaxe, but there are a number of different aspects to understand, so I'm not aware of any single block of code to do it all.
Firstly, the I2C probably uses a dumb "expander" chip, which gives 4 data lines and 4 control lines (including one to the Backlight). Thus the LCD controller needs to be "initialised" and then operated in its 2 cycles of 4-bit wide data mode. That is well-described by Inglewoodpete
HERE. But instead of sending the signals to pins, you would build up a sequence of bytes to send via the I2C Bus.
Then you will probably need to write subroutines to convert the data/control strings that you may have used with the Orbital LCD (or any other intelligent serial display) to simple sequences of control/data bytes (perhaps using the LOOKUP command). But maybe somebody has a better recall of a PICaxe "Finished (and documented) Project" than I do.
EDIT: There are other threads on the topic, but finding them is hindered because the Search engine considers the terms
I2C and
LCD to be "Too short or too common" to search for them.
However, I did find
THIS THREAD with a similar OP question.
Cheers, Alan.