fm's released library has support for using 4 bit mode, i2c, and shift registers using 2 or 3 wires
to talk to a HD44780 based LCD.
i2c is not a generic interface for an LCD. (There are different ways to use I2C to talk to an LCD)
The i2c support in the released library is for using a PCF8574 i2c chip to drive the LCD.
Arduinakias, the LCD & module you have appears to be using a i2c backpack,
the ByVac BV4218.www.byvac.com/downloads/datasheets/BV4218%20DataSheet.pdf
While it does have an i2c interface on it, it does not work the same
as when a PCF8574 chip is driving the LCD.
So unfortunately, fm's currently released newliquidcrystal library does not have support
for the BV4218.
Last summer (2012) ByVac added support for the BV4218 to fm's library.
There were some issues with the early code (it broke the library), but it looks
like the code was updated in sept of 2012.
I glanced at it and for sure it won't compile with pre 1.x Arduino core.
So not sure if it works with Arduino 1.x, but it is probably worth trying it.
Since fm has not created a release since this code was added,
there is no download zip file that includes the 4218 support.
To get the code you will have to go to the download section, then go to the [Tags] tab
and then download the "tip" tag.
The direct link to the zip file for that is here:https://bitbucket.org/fmalpartida/new-liquidcrystal/get/tip.zip
To install and use it is just like the other downloads.
You have to move the existing liquidcrystal library directory off somewhere else.
Then extract the files in the zip image and place it in the liquidcrystal library directory.