MOSFET started to smoke when switching a 12V bulb - how can I troubleshoot the cause?
The 2SK1363 is a fairly inappropriate choice for this application. Look at \$R_{DS(ON)}\$ for instance: -

You have a 30 watt 12 volt lamp and it will try and take 2.5 amps. That current through a fully-activated  2SK1363 means a power dissipation in the MOSFET of 6.875 watts (this is based on typical \$R_{DS(ON)}\$ = 1.1 Ω).
That would need a heatsink for sure. But it's going to be worse than this because to adequately turn on the MOSFET you need a gate drive of at least 10 volts: -

Pick a MOSFET that has a much lower \$R_{DS(ON)}\$ that can be achieved with logic level signals appropriate to the Arduino you are using. For instance, the IRFZ44 (a long-standing favourite of mine) has this useful graph in its data sheet: -

With a gate-source drive level of 4.5 volts and a load current of 2.5 amps, the IRFZ44 will &quot;drop&quot; about 170 mV. That's a power dissipation of 2.5 x 0.17 = 0.425 watts and more appropriate for something like this however, I'd still try and find something that is a tad better because these are typical values.