First, there are inaccuracies and misconceptions in the article you quote, which also relates to obese people with insulin-resistance (pre type II diabetics) rather than normal people. These are:
- I’m not sure what a “low glucose state” is supposed to mean, but the tissues of the organism co-operate to ensure that the blood glucose concentration normally never falls below about 4 mM, with a range of approx. 4–8 mM. Thus, glucose is available in the blood for tissues that can take it up in a particular circumstance. These are not “paltry levels”, as quoted.
- The body’s metabolism cannot predict the future. The scientific credentials of anyone who writes “the body senses that dietary glucose might not be coming anytime soon” are hardly to be trusted. (I don't think much of his literary style either.)
- In short-term starvation, when the blood glucose concentration falls towards the lower end of its range, insulin resistance is not triggered. Insulin secretion from the pancreas stops (and glucagon secretion starts).
- The brain is not sensitive to insulin (i.e. its glucose
transporter(s) is not), whereas the muscle is. This is what allows the brain continued access to glucose.
Some of this is elaborated in my answers to: “What is the purpose of gluconeogenesis?”, “How do muscle cells synthesize glycogen?” and “During starvation, does the human body do anything to prioritize which organs receive nutrients?”. A section from Berg et al. may also be of interest.
Lactate can (and must) be generated by non-aerobic muscle metabolism (i.e. in muscle lacking mitochondria) during exercise because anaerobic glycolysis, with conversion of pyruvate to lactate, is the only way the muscle can obtain ATP to power its contraction. The source of this is the glycogen stores laid down in the muscle after a meal, and glucose taken up from the blood.
I don’t know anything about low-carb diets, but I assume they include protein, so that some of the amino acids from this can be converted to glucose by the liver (glucogenic amino acids) as well as those that cannot (the ketogenic ones that can give rise to ketone bodies). Even if the blood glucose is at the lower end of its range, in the non-starved state the muscle can take it up from the blood. (The blood glucose will be restored by the so-called Cori Cycle.) Even in starvation, if the muscle is working there will be a concentration difference between blood and muscle glucose that will allow uptake.
I think that this question is provoked by the feeling that the existence of ketosis means that glucose is not available to cells. This, as I hope I have explained, is a misconception, perhaps arising from confusion with the different situation of ketosis in long-term starvation.
Finally, there are other sources of lactate besides muscle, regardless of diet. One is the erythrocytes which lack mitochondria and therefore can only use anaerobic glycolysis to produce the ATP they need for active cation transport etc.