It was never known as "General Motors Company Truck" is was General Motors Truck Company (GMTC)
But I tend to agree it was General Motors Company = GMC in the very beginning. When the company became incorporated the GMC meaning simply changed. My information is straight out of "100 years of GMC" put out by General Motors. Mistakes are just as likely as in any of Holden's literature. I also doubt the original intention was solely for truck naming. Although by this time of the GMC Trade Mark they had plenty of names to choose from. This period of time was certainly tumultuous for William Durant and what was his idea for naming at one point could well have been changed by others (quite likely infact) when he was first ousted from the company.

Finding a copy or record of the original Trade Mark and/or Patent should have a clear answer.
I don't know that I care enough to go searching...