Some of this article is outright wrong. Specifically, that Type 1 Diabetics over-produce glucagon. http://www.diabetesselfmanagem... Also, a common factor in T2D individuals is that their muscles are starting to store fat. Your link fails to account for any of that. Now, as for how it's wrong about T1D (I should know, I am one):
Glucagon is prescribed as an emergency injection for hypoglycemia. If the body of a T1D would be consistently putting out glucagon, then glucagon would be worthless as an emergency injection of it.
Now, that said, T1D react differently when insulin levels are low (which since they are not making any/enough of their own, means not injecting any/enough.) As a result of low insulin, the body stops being able to use the glucose that is available. The body cannot recognize a difference between too little insulin, and too little glucose. As a result, the body starts starving, even though the body is full of glucose. As a result the body starts responding as would any starving person: upregulation of glucagon to have the liver produce more glucose, to meet the body's needs.
But remember, the body already has plenty of glucose, just no insulin to make use of it. So thus, the liver just keeps dumping more and more glucose into the system, which isn't being used, and it builds up. At the same time, it's making ketones for other fuel. As the glucose and ketone levels rise, it starts to make the body acidic, which is buffered by bicarb, until the bicarb stores are depleted, and then the body starts going in to acidosis.
There are some T1D who regulate their glucose levels with a ketogenic diet. After the glycogen levels in the liver are depleted, it can't raise the BG in response to starvation, which leads to an easier to manage BG level. However, for these individuals, glucagon injections will not help them with a hypo. But since their body normally has largely switched over to ketone metabolism, they're unlikely to experience hypos anyways.