I've took some more samples and noticed that as the snow blower starts up, the signal is different for about 2 seconds. There are several places where the "normal" signal (the one we have been replicating with an Arduino thus far) are mixed into the starting signal, but the starting signal is predominantly large pulses (in the ms range, not the us range). This may be a coincidence, but there are exactly 41 of the larger pulses, the same number of us range pulses in the "normal" signal. I was going to use my scope's built in function generator to try to feed the signal back into the snow blower while actually using the DeWalt battery, but my function generator's range is 2.5v to -2.5v, so that will not work. I'll take a couple of more samples tomorrow and see if the staring signal is the same each time. If it is I'll try to duplicate it with the Arduino. I've attached a waveform that shows this if anyone else cares to analyze it.
I took more samples and never saw the "larger pulses" I mentioned above again, so I guess that was fluke generated by the GW battery? The starting signal is slightly different though. From here on I'll refer to a "pulse group" as a series of pulses separated by a large (relatively speaking) delay. The first pulse group always happens 1.1 seconds after the Omega pin is pulled high and only has 40 pulses, not 41. The first 5 pulse groups occur with only a 90ms delay between them, where all the pulse groups after than have a roughly 190ms delay between them. I duplicated this, but the snow blower still only runs for 3 seconds. Next, the first 5 pulse groups are have slightly different delays between the pulses. I'm getting ready to duplicate that now.
I also spliced in the Arduino while using the GW battery so the Arduino was generating the signal for the Omega pin instead of the battery. This tells me that the minor voltage difference between the DeWalt battery and the GW battery is no the cause of the problem.
I deleted the original wave I attached and I've attached a new wave recorded from my Rigol MSO5074 which show the start sequence and several seconds of the tool running correctly.