Good find on the Ic being a "continuous" rating. So it's more of a guideline on what to avoid long term rather than a strict limit.
I'm not sure what the safe-operation chart is exactly showing. The times shown are quite short. But again is supports the idea of going over 6A for short pulses.
Wattage is not usually a concern in a device used this way because the pulses are short and the device is used in "saturation". By device "used this way", in our case, I mean an ignitor that is a simply "on" or "off" like a switch (aka "in saturation" or "in cutoff"). We would worry about heat if the device was a current-limiting device (aka "active").
I included a bunch of detail below to explain the reasons why, for anyone interested, but it's not essential.
Let's consider a "perfect" switch as a resistor that can be at infinite ohms, or 0 ohms. A transistor used as a switch is less than perfect. It can't quite be infinite, and it can't quite be 0. Notice in the english pdf it is described as having a low saturation output voltage. That is describing the condition of the transistor trying to be 0 ohms mode. Because it is not at o ohms, the voltage will not be pulled down to exactly 0 volts.
Darlington pairs, due to the "stacked" nature of the design has a drawback. That is that the device does not work as well as a "normal" transistor when acting like a switch. A normal transistor can act more like a short circuit (resistor with resistance close to 0). The darlington is slightly further away from 0. So that is one parameter that is always strived to be improved in the deigns... which is why it's advertised. But it's still not as good as a normal transistor.
The reason I mention all this is because this slight deviation from being a 0 ohm resistor is what causes the heat generation. When the transistor is turned on (acting like a closed switch), the voltage at the negative end of the coil will be close to 0v, but not quite. It is dependent on the current. More current means higher voltage drop.
Let's say the current is 6 amps.
Let's say the transistor can pull the voltage down to about 1.5 volts.
So that is 6 x 1.5 = 9 watts. But the average wattage is what generates overall heat. So let's assume a 50% duty cycle. That drops it to just 4.5 watts. So we are pretty safely in the 40w limit. However, that 40w limit is sometimes specified with a big heatsink and a fan blowing on it. So there is a lot of interpretation you have to take. Like you mention, being in a box does make a big difference. The aluminum plate on the back was a good idea.
I assume Kawasaki did many heat-related tests on the ignitor and they tend to way over-build their electronics.
This was a very long-winded explanation of the wattage factor considering it's often not a consideration. Well the reason for all the detail is that wattage quickly comes to the top of the concerns when we talk about current limiting functions.
So in current limit mode, that switch model changes. When off the resistor is infinite, then when initially turned on, it's 0 ohms (saturation mode). But when we go to current limit mode, it suddenly jumps up to a larger value. That is called "active" mode. That is the mode when wattage becomes the utmost concern.
Let's say the coil is .5 ohm. Let's say we have 14v. Let's say we want to limit the current to 6 amps. What size resistor do we need the transistor to act like?
At 14v, and 6A, we need the total resistance to be 14/6 = 2.33 ohms. So we need a 1.83 ohm resistor in series with the coil. The transistor is that resistor. So what is the wattage on that resistor? It's current squared times the resistance. 6 x 6 x 1.83 = 65.88 watts. With a worst case duty of 50%, it drops to 32.94 watts. That is much higher than the 4.5 watts we had for the non-current limit example.
So it would still be in the spec range long term, but you are starting to creep up on the limit. I will say, in the box with no cooling fan I would not run that close to the wattage limit. It will be smoking hot. Plus you have two transistors side by side there so the actual heat dissipation will have to be doubled. Ignitors running without current limit don't really get hot. In current limit mode, they get untouchably hot.