I believe that Dr. Palmer takes the position that however you can do it, the important things are to lower insulin levels and to keep the body in ketosis, and it doesn’t matter how you achieve those goals.
That said, however, a vegan keto diet is indeed possible, but it is tricky to implement and will require supplementing with certain vitamins. A vegetarian keto diet is easier, especially if one is willing to eat fish, eggs, and dairy. A keto diet based on meat is the easiest to manage.
The reason is that our protein intake is limited by how well the amino acid profile of our meals matches our needs. There are nine essential amino acids the body cannot make for itself. If you get 120% of the amount you require of eight of those amino acids, but only 80% of the ninth amino acid, then only 80% of the protein you are eating will be available to you; the rest will go to waste, since the human body has a very limited capacity for storing amino acids (this is why we need some protein pretty much every day).
The official U.S. dietary guidelines advise getting a certain amount of “high-quality reference protein,” by which they mean beef, since beef contains all the essential amino acids in all the correct proportions. The quality of other protein sources depends on how well they match the amino acid profile of beef.
That said, there is a certain prejudice against meat in our culture. The modern-day vegan movement is rooted in the teachings of Ellen G. White, the Prophetess of the Seventh-Day Adventist church, who believed that meat-eating stimulates lustful thoughts and leads to masturbation. There are also ethical and environmental claims made to support such a diet, but they fall apart under scrutiny. The medical claims that meat is bad for us are based on faulty science, as you will learn if you watch the lecture by Dr. Georgia Ede linked above.
Given that there is an abundance of evidence to suggest that the human race evolved over two million years of meat-eating, for myself I consider it bonkers to suggest that meat can be harmful to our health. Since the Seventh-Day Adventists started the American Dieticians’ Association and the American College of Lifestyle Medicine, however, it is not all that surprising that those organisations’ advice aligns with their founders’ prejudice against meat.
There is a growing movement of people who eat nothing but meat, many of whom have been on a carnivore diet for well over a decade. I mention this not to suggest that you necessarily need to give up plants for the sake of your mental health (though you might well come to find it beneficial), but simply to show that an all-meat diet has no deleterious effects on long-term health.