There is at least one book called "The Keto Diet".

All ketogenic diets are LCHF and those LCHF diets with low enough carbs that don't knock you out of ketosis are ketogenic. This is the only way of eating that works for me to lose and maintain.

Some LCHF diets involve just enough carbs to keep you out of ketosis all the time because "the experts" think ketosis is unhealthy. They don't work for me.

Personally I eat ketogenically most of the time, but occasionally splurge with a few more carbs, but cannot do that all the time because it starts triggering hunger, cravings & binging, so I return to ketogenic eating.

I suspect people started using "LCHF" to emphasize that their diet is not high protein and does not require ketone-level measuring. But I've heard one Ornish-type use the acronym for Low-Calorie High-Fiber, probably to ride the bandwagon. Also, ignorant doctors think ketosis is the same as ketoacidosis, so avoiding the use of "keto" may be helpful to get them to even look at a diet. But "LC" is too broad - some think it can be 40, 60 or 100g of net carbs, but it depends on the person. For me 30g net carbs is my threshold for disaster, so I aim for 20.