Yes, the Xciting 500 does have cold start issues. (2009 Xciting Ri ABS here).
It is due to the drop in battery voltage when turning the engine over; your battery needs to be in good shape, or it will spin but not start up. This is in stark contrast to the Xciting 250, which can still start with a pretty well run-down battery. The XC500 electronics system kicks the bucket under a certain voltage; it needs to have gas and spark near TDC, which is when the starter is using the most current (to compress the air/gas mixture) which makes the battery voltage drop. If the battery loses its cranking capacity, the Xciting will happily turn the engine over and over until the battery is dead without ever starting up. If you lose the time on your clock while turning the engine over, your battery needs to be recharged, and replaced if recharging won't help the situation. Fuel injection happens during the air intake part of the cycle, and that's when the injectors are used, so fuel injection probably doesn't meaningfully contribute to the starting problem. Colder temperatures reduce the available cold cranking current on a battery, which is why the XC500 has (a somewhat deserved) reputation for cold starting issues; the reduction in battery starting capacity can cause the voltage to drop further, crossing that magic line where it simply won't start.
In any event, when you turn the ignition on, WAIT for the engine light to go off before cranking over. When you turn on the ignition, it momentarily turns on the fuel pump to pressurize the fuel rail; you don't want to crank it over while the fuel pump is still running (since it will be an additional load on the battery and drop the voltage more).