Can an arduino tell between being powered by USB or by the jack?
No, you can't do it without extra hardware.

You need to pass VIN through a voltage divider so it sits within 0-5V when powered by 12V, then feed that into an ADC input. Protect the on-board voltage regulator with a diode between 5V (anode) and VIN (cathode). Then you can measure the voltage on VIN and make a decision on if it is around 4.5V or 11.5V

A 22K/10K voltage divider will reduce 12V down to 3.75V, which gives plenty of "headroom" for over-voltage situations (where your 12V unregulated power supply actually provides, say, 15V):