Li-ion (Lithium-ion) are a collective term for all Lithium based batteries.
I'm not sure, but I think your battery have 12 LiFePO4 cells, which means that the correct charging voltage should be around 43.2V to 43.8V
That's true, but very often the term Li-ion is used to describe a LiCoO2 cell with liquid electrolyte, and LiPo describes a cell of the same chemistry with polymer electrolyte.
My experience with other LiFe cells (mostly A123) has been that the cell voltage immediately after full charge is 3.6V and over the next day or so the cell self-discharges to about 3.3V and stays there. I charged the 36V Golden Motor battery and measured the voltage at the charging jack at 40.9V after about 6 hours after charge. That would seem to confirm that there are 12 cells in the pack as you suggest, that have self-discharged so far to about 3.4 V per cell.
The output of the charger is labelled as 42.0 V so yes, it is appropriate for a LiFe pack of 12 cells.
However, I have another problem with this new battery pack. The voltage at the main output is zero, regardless of the switched position of the key switch.
Am I missing something, or do I have a faulty new battery pack?